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AN 


ELEMENTARY    TREATISE 


ON 


MECHANICS. 


TRANSLATED  FROM   THE  FRENCH  OF  M.  BOUCHARLAT. 


/a)DITIONS  AND  EMENDATIONS,  DESIGNED  TO  ADAPT  IT  TO  THE  USE  OF 
THE  CADETS  OF  THE  U.  S.  MILITARY  ACADEMY. 


BY    EDWARD    H.    COURTENAY, 

PROFESSOR  OF  NATURAL  AND   EXPERIMENTAL    PHILOSOPHY  IN  THE  ACADEMY. 


NEW-YORK; 

PRINTED  AND  PUBLISHED  BY  J.  &  J.  HARPER, 

NO   82   CLIFF-STREKT, 

AND   SOLD   BY   THE   BOOKSELLERS   GENERALLY   THROUGHOUT   THE 
UNITED    STATES. 


^^i  ^  Mechanical  Ensine^, 


[Entered  according  to  the  Act  of  Congress,  in  the  year  1833,  by  J.  &  J.  Harper, 
in  the  Office  of  the  Clerk  of  the  Southern  District  of  New-York.] 


•  "^^TkO  .00810/.  ^«^^'■''' 


Es^'neeriBg 
Libiaiy 


PREFACE. 


In  preparing  a  translation  of  Boucharlat's  Elements  of 
Mechanics,  it  has  been  the  principal  object  of  the  translator  to 
supply  a  suitable  text-book  for  the  use  of  the  Cadets  of  the 
United  States'  Military  Academy.  To  accomplish  this  object 
more  effectually,  it  has  been  deemed  necessary  to  introduce 
several  subjects  which  are  not  noticed  in  the  original,  and  to 
extend  or  modify  others,  where  the  methods  of  investigation 
adopted  by  the  Author  appeared  incomplete  or  obscure.  It 
was  also  judged  proper  to  omit  one  or  two  subjects,  the  dis- 
cussion of  which  is  usually  reserved  for  works  of  a  less 
elementary  character. 

These  alterations  were  adopted  with  less  hesitation  as  the 
work  was  principally  designed  for  a  special  purpose  ;  but  it 
is  believed  that  they  will  render  the  work  more  generally 
useful,  by  facilitating  the  comprehension  of  many  of  the  more 
difficult  investigations,  and  by  affording  much  valuable  in- 
formation in  relation  to  those  subjects  which  were  not  dis- 
cussed in  the  original,  but  which  are  generally  admitted  to 
form  an  essential  part  of  an  elementary  course  of  Mechanics. 

In  supplying  the  deficiencies  of  the  original,  reference  has 
been  had  most  frequently  to  the  works  of  Poisson,  Francœur, 
Navier,  Persy,  Genieys,  and  Gregory  ;  and  in  some  few  in- 
stances, the  methods  of  investigation  pursued  by  those  authors 
have  been  adopted  with  but  slight  alterations. 

The  works  of  Boucharlat  have  long  enjoyed  an  unusual 

share  of  public  favour  ;  and  the  hope  is  therefore  entertained 

A2 


4  PREFACE. 

that  the  treatise  now  presented,  in  our  own  language,  will 
prove  a  useful  introduction  to  the  study  of  the  higher 
branches  of  Mechanics,  and  that  it  will  be  received  with  in- 
dulgence by  all  those  who  are  disposed  to  cultivate  a  taste  for 
the  most  interesting  apphcation  of  Mathematical  Science. 

As  the  entire  work  may  be  found  to  constitute  too  exten- 
sive a  course  for  those  students  who  can  devote  but  a  limited 
time  to  the  study  of  Mechanics,  it  was  thought  expedient  to 
indicate  such  of  the  more  difficult  subjects  as  might  be 
omitted.  These  subjects  are  designated  in  the  table  of  con- 
tents by  being  printed  in  italics  ;  and  they  will  be  found  to 
be  unnecessary  in  enabling  the  student  to  comprehend  those 
which  follow. 


CONTENTS. 


PART  FIRST. 

STATICS. 

Introductory  Remarks  and  Definitions 9 

Of  the  Composition  and  Decomposition  of  Forces     ....  II 
Of  Forces  situated  in  the  same  Plane  and  applied  to  a  single 

Point 20 

General  Remarks  on  Forces  situated  in  any  manner  in  Space  24 

Of  Forces  situated  in  Space  and  applied  to  a  Point    ....  27 
Of  the  conditions  of  Equilibrium  of  a  Point  acted  upon  by  several 
Forces,  and  subjected  to  the  condition  of  remaining  upon  a 

given  Surface 31 

Of  the  conditions  of  Equilibrium  of  a  Point  acted  on  by  several 
Forces,  and  subjected  to  the  condition  of  remaining  con- 
stantly on  two  curved  Surfaces,  or  on  a  Curve  of  double 

Curvature 36 

Of  Parallel  Forces 39 

Of  Forces  situated  in  the  same  Plane  and  applied  to  Points  con- 
nected together  in  an  invariable  manner 48 

Of  Forces  acting  in  any  manner  in  Space 60 

Theory  of  the  Principal  Plane,  and  Analogy  existing  between 

Projections  and  Moments 71 

Centre  of  Gravity 79 

Of  the  Centrobaryc  Method 97 

Machines — Cords 99 

Of  the  Catenary 103 

Of  the  Lever 109 

Of  the  Pulley 116 

Of  the  Wheel  and  Axle 120 

Of  the  Inclined  Plane 126 

Of  the  Screw 128 

Of  the  Wedge .131 


6  CONTENTS. 

Page 

Friction 132 

Effects  of  Friction  in  certain  Machines 136 

Of  the  Stiffness  of  Cordage 142 

On  the  Resistance  of  Solids 145 

Of  the  Resistance  to  Compression  or  Extension 147 

Of  the  Resistance  of  a  Solid  to  Flexure  and  Fracture  produced 
hy  a  Force  acting  at  right  angles  to  the  direction. if  the 

Fibres ,. 150 

Of  the  Figure  of  the  Solid  after  Flexure 164 

Of  Solids  of  equal  Resistance  , 175 

Of  the  Principle  of  Virtual  Velocities 177 

Of  the  Position  of  the  Centre  of  Gravity  of  a  System  when  in 

!         Equilibrio 184 


PART  SECOND. 

DYNAMICS. 

Of  the  Law  of  Inertia 187 

Of  uaiform  rectilinear  Motion 188 

Of  varied  Motion , 190 

Of  uniformly  varied  Motion 194 

Of  the  Motion  of  a  Body  projected  vertically  upward  .  .  .197 
Of  the  vertical  Motion  of  a  Bod/  when  acted  upon  by  the 

Force  of  Gravity  considered  as  variable 199 

Of  the  vertical  Motion  of  a  Body  in  a  resisting  Medium  ,     .     .  202 

Of  the  Motions  of  Bodies  upon  Inclined  Planes 205 

Of  curvihnear  Motion 208 

Of  the  Motion  of  a  material  Point  when  compelled  to  describe 

a  particular  Curve  . 221 

Of  the  Motion  of  a  rhaterial  Point  when  compelled  to  move  upon 

a  curved  Surface  .     .     . 229 

Of  the  Motion  of  a  material  Point  on  the  Arc  of  a  Cycloid  .     .  236 

Of  oscillatory  Motion     .     .     .     .     .     .     . 238 

Of  the  Simple  Pendulum     .....     . 240 

Of  the  Centrifugal  Force    .     .     ...     .     .     .     .     .     ,     .  247 

Of  the  System  of  the  World 253 

Of  the  Motions  of  Projectiles  in  Vacuo  ........  272 

Of  the  Motions  of  Projectiles  in  a  resisting  Medium  ....  278 

Of  the  different  Methods  of  measuring  the  Effects  of  Forces  288 


CONTENTS.  f 

Page 

Of  the  direct  Impact  of  Bodies 293 

Of  the  direct  Impact  of  unelastic  Bodies     .......  293 

Of  the  direct  Impact  of  elastic  Bodies 294- 

Of  the  Preservation  of  the  Motion  of  the  Centre  of  Gravity  in 

the  Impact  of  Bodies 297 

Of  the  Preservation  of  living  Forces  in  the  Impact  of  elastic 
Bodies — Relative  velocity  before  and  after  Impact — Loss 
of  living  Force  in  the  Collision  of  unelastic  Bodies  .     .     .  299 

Principle  of  D'Alembert 301 

Of  the  Motion  of  a  Body  about  a  fixed  Axis 309 

Of  the  Moment  of  Inertia 316 

Of  the  Motion  of  a  Body  about  a  fixed  Axis  when  acted  upon 

by  incessant  Forces 320 

Of  the  Compound  Pendulum 322 

Of  the  Motions  of  a  Body  in  Space  when  acted  upon  by  im- 
pulsive Forces 328 

Of  the  Motions  of  a  System  in  Space  when  acted  upon. by  in- 
cessant Forces 333 

General  Equations  of  the  Motions  of  a  System  of  Bodies  .     .  388 
General  Principle  of  the  Preservation  of  the  Motion  of  the 

Centre  of  Gravity  ...     ; 345 


PART  THIRD. 

HYDROSTATICS. 

Of  the  Pressure  of  Fluids 349 

General  Equations  of  the  Equilibrium  of  Fluids 351 

Application  of  the  general  Equations  of  Equilibrium  to  incom- 
pressible Fluids 354 

Application  of  the  general  Equations  of  Equilibrium  to  elastic 

Fluids 359 

Of  the  Pressure  of  heavy  Fluids 36  J 

Of  the    Equilibrium,   Stability,   and    Oscillations   of  floating 

Bodies 366 

Specific  Gravity — Hydrostatic  Balance — Hydrometer     .     .     .  379 

Of  the  Pressure  and  Elasticity  of  Atmospheric  Air  ....  384 

Of  Pumps  for  raising  Water 386 

Of  the  Air-pump 391 

Of  the  Barometer 398 


b  CONTENTS. 

PART  FOURTH. 

HYDRODYNAMICS. 

Page 
Of  the  Discharge  of  Fhiids  through  horizontal  Orifices  .  .  .  407 
Of  the  Motion  of  Water  in  Pipes 429 


I 


i 


ELEMENTS  OF  MECHANICS. 


PART    FIRST. 


STATICS. 

INTRODUCTORY  REMARKS  AND  DEFINITIONS. 

1.  Mechanics  is  the  science  which  treats  of  the  laws  of 
equihbrium  and  motion.  Wlien  appUed  to  solid  bodies  it  is 
divided  into  Statics  and  Dynamics  ;  the  former  discussing 
the  conditions  of  their  equihbrium,  and  the  latter  those  of 
their  motion.  In  the  application  of  Mechanics  to  the  consid- 
eration of  fluid  substances  a  similar  division  is  likewise  made, 
viz.  Hydrostatics,  which  treats  of  the  equilibrium  of  fluids, 
and  Hydrodynamics,  which  investigates  the  circumstances 
resulting  from  their  motions. 

2.  The  object  proposed  in  Statics  being  the  determination 
of  the  laws  of  equilibrium,  this  state  of  equilibrium  may 
always  be  regarded  as  resulting  from  the  mutual  destruction 
of  several  forces. 

3.  The  term  force  or  power  is  applied  to  every  cause  which 
impresses  on  a  body  or  a  material  point  a  motion  or  tendency 
to  motion. 

4.  A  force  may  act  on  a  material  point  either  by  drawing 
the  point  towards  it,  or  by  pushing  the  point  in  advance  of 
it.  The  first  hypothesis  will  always  be  adopted,  unless  the 
contrary  is  expressly  indicated. 

5.  A  material  point  being  solicited  by  a  single  force  will 
naturally  move  in  a  right  line,  since  there  can  be  no  reason 
why  it  should  deviate  to  the  right  rather  than  to  the  left  of 
this  line. 


10  STATICS. 

6.  The  right  line  along  which  a  force  acts  is  called  the  line 
of  direction, 

7.  The  effect  of  a  force  depends,  1°.  On  its  intensity  ;  2°. 
On  its  point  of  application  ;  3°.  On  its  line  of  direction  ;  and 
4°,  On  its  pushing  or  pulling  along  this  line. 

8.  By  the  intensity  of  a  force,  we  understand  its  greater  or 
less  capacity  to  produce  motion. 

9.  If  two  forces  directly  opposed  to  each  other  sustain  in 
equilibrio  a  material  point  or  an  inflexible  right  line,  the  in- 
tensity of  either  one  of  these  forces  may  be  assumed  arbitra- 
rily, provided  we  assign  an  equal  intensity  to  the  second  force. 
A  similar  remark  is  equally  applicable  to  a  system  composed 
of  any  number  of  forces  ;  and  hence  it  appears  that  the  con- 
ditions of  equilibrium  will  depend  simply  on  the  ratios  of  the 
forces,  and  not  on  their  absolute  intensities. 

10.  Having  assumed  one  force  as  a  unit  of  measure,  we 
say  that  a  second  force  is  equal  to  it,  when,  if  directly  op- 
posed to  it, an  equilibrium  would  ensue. 

Two  equal  forces  applied  to  a  material  point,  acting  along 
the  same  right  line,  and  in  the  same  direction,  constitute  a 
double  force  :  in  like  manner  a  triple  force  may  be  regarded 
as  resulting  from  the  union  of  three  equal  forces,  &.c.  ;  so  that 
the  number  of  these  equal  forces  will  constantly  be  propor- 
tional to  their  joint  intensity.  It  may  hence  be  inferred  that 
if  several  forces  solicit  the  point  M  {Fig.  1)  in  the  same  line 
of  direction  from  M  towards  B,  we  can  add  into  one  sum  all 
these  forces,  since  their  joint  effect  will  be  precisely  the  same 
as  that  of  a  single  force  equal  to  their  sum.  For  the  same 
reason  we  should  subtract  from  this  sum,  or  we  should  regard 
as  negative  all  the  forces  which  tend  to  solicit  the  point  from 
M  towards  A. 

11.  The  unit  efforce  being  arbitrary,  it  may  be  repre- 
sented by  any  portion  of  its  line  of  direction. 

12.  When  a  force  is  applied  to  any  point  of  a  body  whose 
several  parts  are  firmly  connected  together,  this  point  cannot 
be  put  in  motion  without  communicating  the  motion  to  the 
other  parts  of  the  body  ;  if,  therefore,  a  force  be  applied  to  any 
point  A  [Fig.  I),  it  will  have  the  same  effect  as  though  it 
were  applied   to  £m.y  other  point  M,  assumed  on  the  line  of 


COMPOSITION   OF    FORCES.  11 

direction  AB,  Moreover,  if  we  drop  the  consideration  of  a 
body,  we  may  still  regard  the  points  in  space  situated  on  the 
line  of  direction  as  mathematical  points,  no  one  of  which  can 
be  moved  without  imparting  its  motion  to  all  the  others. 

13.  It  appears  from  Art.  12,  that  by  interposing  a  fixed 
obstacle  on  the  line  of  direction  of  a  force,  its  effect  will  be 
entirely  overcome. 

14.  Two  equal  forces  P  and  Q,  applied  to  the  points  A  and 
B  of  an  inflexible  right  line  {Pigs.  2  and  3),  and  acting  along 
this  line,  but  in  contrary  directions,  will  sustain  each  other  in 
equilibrio  :  for  if  the  force  P  tends  to  draw  the  point  A  from 
A  towards  a,  the  point  B,  which  is  firmly  connected  with  A 
by  means  of  the  intermediate  points,  will  have  a  tendency  to 
describe  the  space  Bb,  equal  to  Aa  ;  but  by  hypothesis,  the 
force  Q,  tends  to  move  the  point  B  over  a  space  Bb'  equal 
to  Aa  ;  and  since  B  cannot  yield  to  one  of  these  influences 
rather  than  to  the  other,  it  must  remain  immoveable,  and  aji 
equilibrium  will  necessarily  ensue  (Art.  13).  In  like  manner, 
if  the  forces  P  and  Q,  had  been  supposed  to  exert  a  tendency 
to  push  A  and  B,  the  same  consequences  might  have  been 
deduced. 

15.  When  the  right  line  AB  is  reduced  to  a  point,  the  two 
equal  forces,  being  directly  opposed,  are  still  in  equilibrio  ;  but 
if  the  forces  are  unequal,  the  point  M  {Fig.  1)  will  be  moved 
in  the  direction  of  the  greater,  by  a  force  equal  to  the  differ- 
ence of  the  two  unequal  forces. 

Of  the  Composition  and  Decomposition  of  Forces  applied 
to  a  Point. 

16.  When  two  forces  act  upon  a  moveable  point  in  direc- 
tions forming  with  each  other  an  angle  whose  summit  is  the 
point  of  application,  the  state  of  equilibrium  cannot  subsist. 

For,  if  we  suppose  the  two  forces  P  and  Q,  {Fig.  4)  to  be 
in  equilibrio,  we  may  introduce  a  third  force  P'  equal  and 
directly  opposed  to  the  force  P.  The  forces  P  and  d  being 
supposed  to  destroy  each  other,  the  force  P'  must  produce  its 
entire  effect,  and  must  consequently  move  the  point  M  in  a 
direction  from  M  towards  P'.     But  P  and  P',  being  equal  and 


12  STATICS. 

directly  opposite,  must  likewise  destroy  each  other,  and  the 
force  Q.  will  therefore  act  as  though  it  were  alone,  soliciting 
the  point  M  in  a  direction  from  M  towards  Q,  ;  and  since  it 
is  impossible  for  the  point  M  to  move  in  two  directions  at 
the  same  time,  we  cannot  suppose  that  P  and  Q,  are  in  equi- 
Ubrio  without  involving  an  absurdity. 

17.  Since  an  equilibrium  cannot  subsist  between  two  forces 
whose  lines  of  direction  are  not  coincident,  the  point  M  will 
tend  to  move  in  a  certain  direction  MR,  as  though  it  were 
solicited  by  a  single  force  R.  This  force  is  called  the  result- 
ant of  the  two  others,  and  the  original  forces  are  called  com- 
ponents. 

It  may  be  observed  that  two  forces  which  have  a  resultant 
do  not  always  intersect.  For  example,  if  two  parallel  forces 
P  and  Q,  be  supposed  to  act  on  a  body,  and  if  a  third  force  R 
be  found  which  shall  produce  the  same  effect,  R  will  be  the 
resultant  of  the  forces  P  and  Q,. 

18.  Having  examined  the  conditions  of  equilibrium  of  two 
equal  forces  acting  on  a  point,  the  most  simple  case  which 
next  presents  itself  is  that  of  three  equal  forces  applied  to  the 
same  point.  Let  P,  Q,  and  R  represent  these  forces  ;  if  they 
produce  an  equilibrium,  their  directions  will  divide  the  cir- 
cumference of  a  circle  whose  centre  coincides  with  the  point 
of  application,  into  three  equal  parts  {Fig.  5)  :  for  since  the 
same  reasons  may  be  adduced  to  prove  that  the  point  should 
tend  to  move  in  the  direction  of  each  of  these  forces,  it  fol- 
lows that  it  cannot  yield  to  the  influence  of  either  in  prefer- 
ence, and  must  consequently  remain  at  rest. 

19.  The  equal  angles  PMa,  PMR,  and  QMR  {Fig.  5), 
being  measured  by  one-third  of  the  entire  circumference,  each 
of  them  is  equal  to  f  of  a  right  angle,  or  120°,  Hence,  if  one 
of  the  three  lines  PM,  QM,  or  RM  be  prolonged  through  M,  it 
will  bisect  the  angle  formed  by  the  other  two.  If  MS,  for  ex- 
ample, be  the  prolongation  of  the  line  RM,  the  angles  PMS, 
CIMS  will  be  equal,  being  supplements  of  the  equal  angles 
PMR  and  GlMR  ;  whence  it  appears  that  MS  bisects  the  angle 
PMGl. 

20.  Let  us  next  suppose  the  two  equal  forces  P  and  Q, 
{Fig.  6)  to  be  applied  perpendicularly  to  the  extremities  A 


COMPOSITION    OP    FORCES,  13 

and  B  of  a  right  line  AB  ;  the  resultant  of  these  forces  will 
pass  through  the  point  O,  the  middle  of  the  line  AB,  and  will 
be  equal  in  intensity  to  the  sum  of  the  intensities  of  the  two 
forces  P  and  Q,.  For,  draw  through  the  points  A  and  B  the 
four  right  lines  AC,  AD,  BC,  BD,  each  forming  with  AB  an 
angle  equal  to  ^  of  a  right  angle  :  the  triangles  ACB,  ADB 
will  be  isosceles,  and  will  have  the  sides  AC,  CB,  AD,  DB 
equal  to  each  other. 

The  right  lines  AB,  CD  will  intersect  each  other  at  right 
angles,  and  the  figure  ACBD  will  be  a  rhombus  :  the  sides 
of  this  rhombus  and  their  prolongations  determine  by  their 
intersections  the  four  obtuse  angles  ACB,  ADB,  P'AC,  Q'BC, 
each  of  which  is  equal  to  |  of  a  right  angle  ;  for,  the  angle 
CAD  being  by;  construction  equal  to  |  of  a  right  angle,  its 
supplement  P'AC  must  be  equal  to  |  of  a  right  angle  ;  and 
since  the  opposite  sides  of  the  rhombus  are  parallel,  the  angle 
ACB  is  equal  to  P'AC,  and  is  consequently  equal  to  f  of  a 
right  angle.  The  same  may  be  proved  of  the  angles  CBQ,' 
and  ADB.  Moreover,  since  the  line  CD  bisects  the  angle 
ACB,  which  was  proved  equal  to  f  of  a  right  angle,  it  follows 
(Art.  19)  that  the  three  angles  ACB,  ACS,  and  BCS  are  equal 
to  each  other.  In  like  manner  it  may  be  shown  that  there 
are  three  equal  angles  at  each  of  the  points  A,  B,  and  D. 

21.  We  will  now  apply  at  the  points  A,  B,  C,  D,  which  are 
supposed  firmly  connected  together,  twelve  equal  forces,  dis- 
tributed as  follows  : 

At  the  point  A  three  equal  forces  P,  P',  P", 
At  the  point  B  three  equal  forces  Q,,  Q,',  Q,", 
At  the  point  C  three  equal  forces  S,  S',  S", 
At  the  point  D  three  equal  forces  V,  V,  V"  ; 
forming  with  each  other  angles  equal  to  |  of  a  right  angle  : 
these  twelve  forces  will  sustain  each  other  in  equilibrio. 

But  the  forces  P'  and  V",  Q'  and  V,  being  equal,  and 
directly  opposed,  will  destroy  each  other,  as  also  will  the  forces 
P"  and  S',  Q,"  and  S".  If,  therefore,  an  equilibrium  is  main- 
tained in  the  system,  it  must  subsist  between  the  four  forces 
P,  Q,  S,  and  V.  The  two  last,  acting  in  the  same  direction 
along  the  line  DC,  are  equivalent  to  a  single  force  equal  to 
their  sum,  which  may  be  applied  at  O,  a  point  in  their  line  of 

2 


14  STATICS. 

direction.  Thus,  an  equilibrium  will  take  place  between  the 
forces  P  and  Q,  and  a  force  R  whose  line  of  direction  passes 
through  the  middle  of  the  line  AB,  and  whose  intensity  is 
equal  to  the  sum  of  the  intensities  of  P  and  Q. 

If  we  suppress  P  and  Q,  the  equilibrium  will  be  destroyed, 
but  it  may  again  be  established  by  applying  at  O  a  single 
force  R'  equal  and  directly  opposed  to  the  force  R.  The  force 
R'  must  therefore  produce  an  effect  precisely  equal  to  the  joint 
effect  of  P  and  Q,  and  will  consequently  be  their  resultant. 
We  hence  infer  that  tlie  resultant  of  ttvo  equal  and  jmrallel 
forces  is  equal  to  their  sum,  is  parallel  to  them,  and  divides 
equally  the  line  AB,  which  is  draivn  j^erpendicular  to  the 
co?nmou  direction  of  those  forces. 

22.  To  determine  the  resultant  of  two  unequal  parallel 
forces  P  and  Gl  applied  to  the  extremities  A  and  B  of  a  right 
line  AB  {Fig.  7),  we  will  suppose  p  to  represent  the  unit  of 
force,  and  make  mp==V,  7ip=Q,.  The  ratio  of  m:n  will  be 
the  same  as  that  of  the  forces  P  and  d.  Let  the  right  line 
AB  be  also  divided  in  the  same  ratio  at  the  point  D,  and  we 
shall  have  the  proportion 

P  :  Gl  :  :  AD  :  DB («). 

On  the  prolongations  of  AB,  take  AA'=AD,  and  BB'=BD  ; 
we  shall  then  have,  since  A'D  and  DB'  are  double  AD  and  DB, 

P  :  a  :  :  A'D  :  DB'  :  :  w  :  n. 
If  then  we  divide  A'D  into  m  equal  parts,  DB'  Avill  contain  n 
such  parts,  and  A'B'  will  contain  one  of  these  parts  as  many 
times  as  p  is  contained  in  P+Q,.  And  since  any  two  of  the 
points  of  division  a',  a",  a'",  6cc.  separate  three  equal  parts,  wh  ile 
three  points  separate  four  parts,  &c.,the  number  of  equal  parts 
in  the  line  A'B'  will  exceed  by  unity  the  number  of  points  of 
division.  A  force  being  applied  at  each  point  of  division, 
there  will  remain  one  of  the  number  m-{-n,  of  which  one  half 
may  be  applied  at  A',  and  the  other  at  B'  ;  the  several  partial 
forces  will  thus  be  distributed  throughout  the  line  A'B'.  But 
the  points  A'  and  D  being  equally  distant  from  the  point  A, 
the  force  ^p  applied  at  A'  may  be  combined  with  one  lialf  of 
the  force  p  applied  at  D,  and  their  resultant,  which  is  equal  to 
their  sum,  will  pass  through  A.  The  same  remarks  will 
apply  to  the  forces  />  and  p  applied  at  a'  and  a^  to  the  forces 


COMPOSITION   OF    FORCES.  15 

p  and  p  applied  at  a"  and  a,,,  &-c.  ;  thus,  the  total  resultant 
of  the  partial  forces  distributed  along  A'D,will  be  equal  to 
their  sum  P,  and  will  pass  through  the  point  A.  In  like 
manner  it  may  be  shown  that  the  forces  applied  to  DB'  may- 
be replaced  by  Q,  ;  and  the  entire  system  of  partial  forces  may 
therefore  be  replaced  by  the  two  forces  P  and  Q,  applied  at 
the  points  A  and  B. 

But  these  parallel  forces  may  be  otherwise  compounded, 
by  combining  them  in  pairs  taken  at  equal  distances  from  the 
middle  point  O  of  the  line  A'B'  ;  and  it  may  thus  be  easily 
shown  that  the  resultant  of  the  whole  system  will  pass 
through  the  point  O,  and  will  be  equal  to  P+Q,. 

The  position  of  the  point  O  must  now  be  determined.  For 
this  purpose,  it  may  be  remarked  that  A'O  {Fig.  7),  being  one- 
half  of  A'B',  is  equal  to  AB  ;  and  by  substituting  this  value 
in  the  equation 

AO=A'0— A'A, 
which  results  immediately  from  an  inspection  of  the  figure, 
we  shall  obtain  AO=AB— AA',  or  AO=AB— AD=DB.  In 
a  similar  manner  it  may  be  shown  that  OB=AD  ;  and  by 
substituting  these  values  of  DB  and  AD  in  the  proportion  (a), 
there  will  result 

Q  :  P  :  :  AO  :  OB {b). 

If  P  and  Gt  are  incommensurable  for  the  unit  p,  this  pro- 
portion which  results  from  the  division  of  A'B'  into  m-{-7i 
equal  parts,  might  seem  to  fail  :  but  by  diminishing  indefi- 
nitely the  value  of  the  unit  p,  and  increasing  in  the  same 
proportion  the  number  of  these  divisions,  the  demonstration 
becomes  applicable  to  all  cases,  since  the  equal  parts  Aa',  a'a", 
&c.  being  indefinitely  small,  the  points  of  division  will  then 
become  continuous. 

23.  This  proposition  is  equally  true  when  the  two  parallel 
forces  P  and  Gl  are  applied  to  the  extremities  of  an  oblique 
line  CD  {Fig:  8).  For,  by  drawing  AB  at  right  angles  to 
the  common  direction  of  the  two  forces,  and  transferring  the 
points  of  application  to  the  points  A  and  B  in  their  lines  of 
direction,  the  proportion  {b)  will  evidently  subsist  ;  but  the 
similar  triangles  AGO,  EDO,  give  AO  :  OB  :  r  OC  :  OD  ; 
whence  we  obtain 


16  STATICS. 

a  :  P  :  :  OC  :  OD  : 

and  we  therefore  infer  that  when  two  parallel  and  unequal 
forces  P  a7id  Q,  are  apjAied  to  the  extremities  of  a  right 
line  CD,  their  residtant  will  divide  this  line  in  the  inverse 
ratio  of  the  intensities  of  tJie  forces. 

24.  By  the  aid  of  this  theorem  we  can  readily  demonstrate 
that  of  the  parallelogram  of  forces,  which  may  be  enun- 
ciated as  follows  : — If  any  two  forces  P  and  Q,  applied  to  a 
"point  A  {F'g-  9)  he  represeiited  in  direction  and  intensity  by 
the  lines  AB  and  AC,  their  I'esultant  ivill  be  represented  in 
direction  and  intensity  by  the  diagonal  of  the  parallelogram 
constructed  upon  the  lines  AB  a7id  AC. 

It  is  immediately  obvious  that  the  resultant  will  pass 
through  the  point  of  application  of  the  forces  ;  since  the 
forces  conspire  to  solicit  this  point,  and  their  resultant,  which 
may  replace  them,  must  therefore  contain  it. 

25.  The  resultant  of  the  two  forces  P  and  Q,  will  likewise 
be  contained  in  the  plane  of  those  forces.  For,  if  it  be  situ- 
ated above  this  plane,  a  position  in  all  respects  similar  can  be 
selected  below  the  plane  :  the  same  arguments  may  then  be 
advanced  to  prove  that  its  direction  coincides  with  either  of 
these  lines  ;  and  since  the  resultant  cannot  have  two  direc- 
tions, we  infer  that  it  coincides  with  neither. 

26.  It  may  also  be  proved  that  the  resultant  of  two  equal 
forces  [Figs,  10,  11,  12)  will  bisect  the  angle  included  between 
them. 

For,  if  we  suppose  Km  to  represent  the  resultant  of  the  two 
forces  P  and  Q,  and  draw  AD  bisecting  the  angle  PAQ.,  a 
line  An  may  always  be  found,  whose  position  with  respect  to 
AD,  AQ,,  and  AP  shall  be  precisely  similar  to  that  of  Am 
with  respect  to  AD,  AP,  and  AGI  ;  hence,  the  same  reasons 
which  would  prove  Am  to  b§  the  resultant,  become  equally 
applicable  to  An,  and  it  might  thence  be  inferred  that  there 
are  two  resultants  :  this  being  impossible,  we  conclude  that 
the  resultant  coincides  with  AD. 

27.  Let  the  two  unequal  forces  P  and  Q.  be  now  supposed  to 
act  upon  the  point  A  [Fig.  13),  and  let  the  parallelogram  ABDC 
be  constructed,  whose  sides  AB  and  AC  are  taken  on  the  lines 
of  direction  of  those  forces,  and  are  proportional  to   theiE- 


COMPOSITION   OF   FORCES.  17 

intensities.  It  has  already  been  shown  that  the  resultant  will 
pass  through  A,  and  it  remains  to  be  proved  that  it  will  also 
pass  through  D,  the  extremity  of  the  diagonal  AD.  Having 
taken  DE=AB=P,*  draw  EF  parallel  to  AB,  and  apply  at 
E  and  F,  in  contrary  directions,  the  two  forces  Q,',  Gl",  each 
equal  to  Q,.  Since  these  forces  will  destroy  each  other,  we 
can  substitute  for  P  and  Q,  the  four  forces  P,  Q,,  CI',  and  Q,". 
But  by  regarding  P  and  Gl'  as  two  parallel  forces  applied  to 
the  extremities  of  an  inflexible  line  BE,  and  having  obtained 
by  construction  the  proportion 

P  :  a'  :  :  DE  :  BD, 
it  follows  immediately  from  the  preceding  theorem,  that  the 
resultant  R  of  P  and  Q,'  will  pass  through  the  point  D; 
Again,  if  we  transfer  the  force  Q,,  and  apply  it  at  F,  in  its  line 
of  direction,  the  two  equal  forces  Q,  and  Q,"  will  have  a  re- 
sultant S,  which,  bisecting  the  angle  Q^FGl",  will  pass  through 
D,  the  opposite  angle  of  the  rhombus  CDEF.  We  thus  ob- 
tain two  forces  R  and  S  which  are  equivalent  to  the  original 
forces  P  and  Gl  ;  aad  since  the  forces  R  and  S  pass  through 
the  point  D,  the  resultant  of  P  and  Gl  will  likewise  pass 
through  the  same  point. 

28.  It  will  now  be  proved  that  if  the  intensities  of  the  forces 
be  represented  by  AB  and  AC,  the  diagonal  AD  will  repre- 
sent the  intensity  of  the  resultant  {Pig.  14); 

If  at  the  point  A  {Mg.  14),  and  in  the  direction  AD  of  the 
diagonal  of  the  parallelogram  constructed  on  the  sides  AB=P, 
AC=Gl,  there  be  applied  a  force  X  equal  and  directly  op- 
posed to  the  resultant  of  P  and  Q,,  an  equilibrium  will  take 
place  between  the  forces  P,  Q,  and  X.  But  we  may  regard 
Q,  as  equal  and  directly  opposed  to  the  resultant  of  the  forces 
P  and  X  ;  hence  it  follows,  that  if  through  the  extremity  B 
of  the  line  AB  a  line  be  drawn  parallel  to  X,  intersecting  at 

*  It  should  be  remarked  that  the  expression  AB=:P  is  merely  intended  as 
an  abridged  method  of  stating  that  the  line  AB  represents  the  relative  intensity 
of  the  force  P,  when  compared  with  the  unit  of  force  whose  intensity  is  likewise 
represented  by  a  line.  In  like  manner,  we  speak  of  the  "  force  AB,"  denoting 
thereby  that  the  line  AB  represents  the  line  of  direction  and  relative  intensity 
of  the  force.    These  abbreviations  have  been  sanctioned  by  usage. 

B 


1 8  STATICS. 

E  the  prolongation  of  the  Une  AC,  which,  as  has  been  already 
shown,  coincides  in  direction  with  the  diagonal  of  the  paral- 
lelogram constructed  on  P  and  X,  the  line  BE,  being  a  side  of 
this  parallelogram,  will  be  equal  to  the  opposite  side,  which 
must  represent  X  :  but  BE,  being  also  the  side  of  the  paral- 
lelogram ED,  is  equal  to  the  opposite  side  AD,  which  repre- 
sents the  diagonal  of  the  parallelogram  constructed  upon  P 
and  Q,  ;  whence  X=AD,  and  the  intensity  of  the  resultant  is 
likewise  measured  by  the  length  of  the  diagonal. 

29.  One  of  the  simplest  corollaries  which  may  be  deduced 
from  the  foregoing  proposition  is  the  trigonometrical  relation 
existing  between  the  components  P  and  Q,  and  their  resultant 
R  {Fig.  15).  To  obtain  this  relation,  we  will  assume  on 
the  directions  of  these  forces  the  parts  AB  and  AC  propor- 
tional to  their  intensities,  and  constructing  the  parallelogram 
ABDC,  we  shall  have  the  proportion 

P  :  a  :  R  :  :  AB  :  AC  :  AD. 
And  from  the  equality  of  the  sides  DD  and  AC,  we  shall  have 
in  the  triangle  ABD, 

P  :  a  :  R  :  :  AB  :  BD  :  AD. 
But  the  proportionality  of  the  sides  of  the  triangle  to  the  sines 
of  tlie  opposite  angles  gives 

AB  :  BD  :  AD  :  :  sin  BDA  :  sin  BAD  :  sin  ABD. 
Hence  we  deduce 

P  :  a  :  R  :  :  sin  BDA,  :  sin  BAD  :  sin  ABD. 
The  determination  of  the  relations  between  P,  Q,  and  R  is 
thus  reduced  to  the  solution  of  a  case  in  plane  trigonometry. 

30.  K  there  be  given,  for  example,  the  two  components 
AB  and  AC,  and  the  angle  BAC  contained  between  them, 
and  it  be  required  from  these  to  determine  the  resultant,  we 
shall  have,  in  the  triangle  ABD,  the  sides  AB,  BD,  and  the 
angle  B  equal  to  the  supplement  of  BAC.  With  these  data  we 
readily  obtain  the  value  of  the  side  AD=R,  by  means  of  the 
formula 

R3  =P2  -j-ds  _  2Pa  COS  B. 
If  in  this  formula  we  wish  to  introduce  the  angle  included 
between  the  two  forces,  since  the  angle  B  is  the  supplement 
of  the  angle  BAD,  we  shall  have  the  relation  cos  B= — cos  Aj 


COMPOSITION   OP   FORCES.  19 

whence  by  substitution  the  following  equation  is  obtained 
between  the  resultant,  the  two  components,  and  the  angle 
included  between  them, 

R2  =P-  +0,2  +2Pa  cos  A (1). 

31.  When  the  angle  A  becomes  equal  to  90°,  the  parallelo- 
gram ABDC  {Fig-.  16)  becomes  a  rectangle,  and  cos  A=0. 
The  general  relation  between  the  resultant  and  its  two  com- 
ponents is  then  reduced  to 

R2=p2+a^ 

The  solution  of  the  converse  problem,  or  the  resolution  of 
a  single  force  R  into  two  components  P  and  d,  having  given 
directions,  is  readily  effected  by  constructing  a  parallelogram 
upon  the  line  representing  the  given  force  as  a  diagonal,  the 
sides  of  the  parallelogram  having  the  directions  of  the  re- 
quired components. 

32.  When  there  are  several  forces  lying  in  different  planes, 
but  all  meeting  in  a  single  point,  the  resultant  of  the  system 
can  always  be  determined  ;  for,  by  combining  these  forces  in 
pairs,  and  substituting  each  resultant  for  its  two  components, 
the  number  of  forces  will  be  successively  reduced,  and  we 
shall  finally  obtain  but  a  single  resultant. 

33.  The  method  of  compounding  any  number  of  forces 
which  has  just  been  explained  gives  rise  to  a  remarkable 
graphic  construction.  Thus,  let  P,  P',  P",  P'",  &c.  represent 
any  forces  whose  directions  intersect  at  the  point  A  {Fig: 
17),  and  whose  intensities  are  expressed  by  the  hues  Ap,  Ap', 
Ap",  Ap"\  &c,  assumed  on  the  respective  lines  of  direction  ; 
through  the  point  p  draw  the  line  pr  parallel  and  equal  to 
the  line  Ap',  and  complete  the. parallelogram  Aprp' ]  the  di- 
agonal Ar=R  will  be  the  resultant  of  the  forces  P  and  P': 
in  like  manner,  by  drawing  rr'  parallel  and  equal  to  Ap",  and 
forming  the  parallelogram  Arr'p",  the  diagonal  Ar'  will  be  the 
resultant  of  R  and  P",  and  therefore  the  resultant  of  the  three 
forces  P,  P',  and  P".  By  continuing  this  process,  a  polygon 
Aprr'r"  would  be  formed,  having  its  sides  parallel  to  the 
directions  of  the  forces,  and  their  lengths  representing  the 
intensities  of  those  forces.  The  distances  from  the  point  A 
to  the  angles  of  this  polygon  will  be 

B2 


20  STATICS. 

Ar=the  resultant  of  P  and  P', 
Ar'=the  resultant  of  P,  P',  and  P", 
A?-"=the  resultant  of  P,  P',  P",  and  P'". 
And  by  repeating  the  construction  for  any  number  of  forces, 
the  distance  from  the  point  A  to  the  extremity  7-^'''  of  the  last 
side  of  the  polygon  will  be  equal  to  the  resultant  of  the  en- 
tire system. 

Of  Forces  situated  m  the  same  Plane,  and  applied  to 
a  single  Point. 

34.  Let  P,  P',  P",  &c.  {Pig.  18)  represent  several  forces 
situated  in  the  same  plane,  their  directions  intersecting  at 
the  point  A  ;  through  this  point  let  there  be  drawn  the  rec- 
tangular axes  Ax  and  Ky  ;  then,  denoting  the  respective 
intensities  of  these  forces  by  AP,  AP',  AP",  (fcc,  let  each  be 
decomposed  into  two  components,  whose  directions  shall 
coincide  with  the  rectangular  axes. 

For  this  purpose  we  will  represent  by  «,  «',  «",  (fee.  the 
angles  included  between  the  forces  and  the  axis  of  a;,  and  by 
/3,  /3',  /3",  (fee.  the  angles  which  they  form  with  the  axis  of  y. 

In  the  right-angled  triangle  ABC  {Fig.  19),  the^  side  AC 
being  expressed  by  AB  cos  A,  and  the  side  BC  by  AB  sin  A, 
the  components  of  the  forces  P,  P',  P",  (fee.  in  the  directions 
of  the  two  axes  are  readily  obtained  :  for  the  force  P  repre- 
sented by  AB,  forming  an  angle  ct,  with  the  axis  of  x,  and  an 
angle  /3  with  the  axis  of  y,  will  have  for  its  components  along 
these  axes, 

AC=P  cos  a,  BC=P  cos  /3. 
In  like  manner,  the  forces  P',  P",  P'",  (fee.  will  have  for  their 
components  in  the  direction  of  Aa:, 

P'  cos  «',  P"  cos  «",  P'"  cos  «'",  (fee, 
and  in  the  direction  of  the  axis  Ay, 

F  cos  /3',  P"  cos  /3",  P'"  cos  /3'",  (fcc. 

If  the  sum  of  the  components  acting  in  the  direction  of  x  be 

taken,  as  also  the  sum  of  those  acting  in  the  direction  of  y, 

we  shall  have,  denoting  these  sums  by  X  and  Y  respectively,. 

P  cos  «+P'  cos  «'-f-P"  cos  «"-f  «fec.=X, 

P  cos  |34-F  cos  |3'+P"  cos  /3"-f  (fec.=Y; 


FORCES    APPLIED   TO   A    POINT.  21 

and  the  entire  system  will  thus  be  reduced  (Art.  10)  to  two 
forces,  of  which  one  X  is  directed  along  the  line  Ax,  the  other 
Y  acting  along  the  line  Ay.  Calling  R  the  resultant  of  these 
two  forces,  its  value  may  be  determined  from  the  equation 

Xa4.Y2=R2. 

35.  For  the  purpose  of  rendering  the  preceding  determi- 
nation of  the  value  of  the  resultant  general,  we  have  attrib- 
uted the  positive  sign  to  all  the  cosines  which  enter  into  the 
expressions  for  X  and  Y  ;  but  it  will  be  necessary  in  practice 
to  regard  the  essential  signs  with  which  these  quantities  are 
severally  affected.  The  following  considerations  will  serve 
to  explain  the  necessity  of  this  distinction.  Let  a  point  M 
{Pig.  20)  be  solicited  by  a  force  represented  in  intensity  by 
the  line  MP.  By  decomposing  this  force  into  two  others 
whose  directions  shall  coincide  with  the  rectangular  axes 
Mx  and  My,  and  calling  «  the  angle  which  the  direction  of 
the  force  makes  with  the  axis  Mx,  its  two  components  will 
evidently  become 

MC=MP  sin  cc,  MD=MP  cos  «. 
The  forces  which  are  directed  in  the  line  Mx,  being  regarded 
as  positive  when  they  act  from  M  towards  x,  the  component 
MD  will  obviously  be  positive.  If  the  force  MP  should  as- 
sume the  position  MP',  the  angle  «  would  be  increased,  and 
its  cosine  diminished  ;  and  if  the  angle  becomes  greater 
than  90°,  the  direction  of  the  force  will  fall  in  the  second 
quadrant.  In  this  case  it  will  assume  the  position  MP",  and 
the  cosine  of  the  angle  will  change  its  sign.  But  it  is  evident 
that  the  component  MD"  of  the  force  MP"  becomes  also  nega- 
tive, since  it  solicits  the  point  M  in  a  direction  opposite  to 
that  in  which  it  was  urged  by  the  component  MD.  Thus 
it  appears  that  the  signs  of  these  two  components  result  from 
the  signs  of  the  cosine  of  a,  and  hence  the  forces  MP,  MP', 
&c.,  which  solicit  a  point,  may  be  always  regarded  as  essen- 
tially positive,  provided  we  attribute  the  appropriate  signs  to 
the  cosines  of  the  angles  which  they  form  with  the  axes. 

36.  If  the  force  under  consideration  fall  below  AB,  as  in 
the  position  MP'",  the  angle  x  being  measured  by  the  arc 
ALBP'",  will  be  greater  than  two  right  angles.  To  avoid 
this  inconvenience,  it  has  been  agreed  to  reckon  the  angles  as 


22  STATICS. 

and  /J  indiscriminately  on  each  side  of  their  respective  axes. 
Thus  when  the  force  falls  beneath  AB,  the  angle  «e  will  be 
measured,  not  by  the  arc  ALBF",  but  by  the  arc  AP'",  which 
has  the  same  cosine.  By  this  arrangement  all  the  arcs  em- 
ployed are  less  than  180^.  It  is  true  that  when  the  angle  «  is 
alone  given,  the  direction  of  the  force  would  appear  inde- 
terminate, since  this  angle  may  be  counted  either  from  A  to 
Pj  or  from  A  to  P'"  ;  but  this  ambiguity  will  immediately  dis- 
appear by  considering  the  value  of  the  angle  /3,  which  is  evi- 
dently acute  for  the  force  MP,  but  obtuse  for  the  force  MP'". 

37.  Whatever  may  be  the  direction  of  the  given  force, 
since  it  must  necessarily  lie  in  one  of  the  four  right  angles 
formed  by  the  axes  around  the  point  M,  its  position  must 
correspond  to  some  one  of  those  given  in  Figs.  21,  22, 23,  24. 

In  the  first  quadrant,»  and  P  being  acute  give  cos«  positive,  cos  P  positive, 

In  the  second, a,  obtuse  and/3  acute  give  cos  «negative,  cos /J  aegBtwe,  - 

In  the  third, «obtuse  and  P  obtuse  give  cosa  negative,  cos  (3  negative, 

In  the  fourth, «acute  and  /3  obtuse  give  cosas  positive,  cos  (}  negative. 

Each  of  these  angles  will  be  less  than  180°. 

38.  It  may  be  observed  that  the  signs  of  these  cosines  are 
similar  to  those  of  the  co-ordinates  x  and  y  of  the  point  B- 
For  example,  if  the  point  be  situated  within  the  angle  .r'Ay 
{Fig.  22),  X  will  be  negative  and  y  positive,  while  at  the  same 
time  we  shall  have  cos  «  negative  and  cos  j3  positive. 

39.  For  the  purpose  of  making  an  application  of  the  pre- 
ceding principles,  let  us  determine  the  resultant  of  the  five 
forces  P,  P'j  P",  P'",  P"j  which  are  situated  as  represented 
in  Fig.  25,  and  solicit  the  point  A.  By  attributing  to  the 
components  of  the  forces  the  positive  or  negative  signs  cor- 
responding to  the  angles  which  are  acute  or  obtuse,  the  com- 
ponents of 

P    ^  C  +P    cos  «,  -fP  cos/3, 

F  -f  P'  cos  «',  -F  cos  (3', 

P"    i  will  be  I   -fP"cos«", -P"cos/3", 

P'"  —  P''"C0S«"',-P"'C0S/3'", 

P-  [    — P"'C0S«"',+P"'C0S/3". 

Having  taken  the  sum  of  the  components  which  act  in  one 
direction,  we  subtract  from  it  the  remaining  components 
which  act  in  an  opposite  direction,  and  we  thus  obtain 


FORCES    APPLIED   TO   A    POINT.  23 

P  COS  «4-P'  COS  «'4-P"  cos  «"-F"  cos  «'"— P^  cos  «"=X, 

P  cos  /3  +  P''  COS  /3"  — P'COS  ^'  — P"  COS  ^"— P'"  COS  /3"'=Y. 

40.  If  we  defer  the  determination  of  the  signs  of  the  cosines 
until  we  wish  to  make  an  appHcation  of  the  preceding  equa- 
tions, the  several  terms  may  be  written  with  the  positive  sign, 
and  the  general  form  of  the  equations  will  then  become 

P  cos  «4-P'  cos  «'+P"cos  «"+&c.==X (2), 

P  cos  /3+F  cos  13' -fP"  cos  Ii"+ÔÙC.=Y (3). 

41.  The  resultant  being  represented  by  the  diagonal  of  a 
rectangle,  the  lengths  of  whose  sides  are  denoted  by  X  and  Y, 
its  value  will  be  determined  by  the  equation 

R=^(X2+Y==) (4). 

The  position  of  the  resultant  remains  to  be  determined.  If 
we  denote  by  a  and  b  the  angles  which  the  resultant  forms 
with  the  co-ordinate  axes,  we  shall  have 

X=R  cos  a,  Y=R  cos  b  ; 
whence 

cos  a=-^,  cos  6=^5- (5). 

K  K 

The  positions  and  intensities  of  the  forces  being  given,  the 
values  of  X  and  Y  may  be  immediately  deduced  from  the 
equations  (2)  and  (3).  These  values  being  substituted  in  the 
equation  (4),  make  known  the  value  of  the  intensity  of  the 
resultant,  and  its  position  may  be  determined  from  the  equa- 
tions (5). 

42.  Its  line  of  direction  passing  through  the  origin  A  {Pig. 

26),  will  liave  for  its  equation 

sin  a 

?/=a;  tang:  «i  or  V=a^ ; 

•^  &    >       y       cos  a  ' 

and  by  substituting  cos  b  for  sin  a,  since  a  and  b  are  com- 
plements of  each  other,  the  equation  becomes 

cos  b 

y~x , 

co^« 

and  by  substituting  in  this  equation  the  values  of  cos  a  and 

cos  b  given  in  equations  (5),  we  have 

Y 

^=x-^*- 

43-  When  an  equilibrium  takes  place,  the  intensity  of  the 


24  STATICS. 

resultant  becomes  equal  to  zero  ;  and  the  formula  (4)  then 
assumes  the  form 

V(X2+Y2)=0,  or  X2  4-Y2=0. 
But  since  every  square  is  essentially  positive,  the  preceding 
equation  cannot  be  true,  unless  each  of  its  terms  is  separately 
equal  to  zero  ;  hence 

X=0,  Y=0. 
Such  are  the  equations  which  express  the  conditions  of  equi- 
librium of  any  number  of  forces  situated  in  the  same  plane, 
and  acting  on  a  point. 

44.  If  X  alone  were  equal  to  zero,  we  should  have 

R=Y,  cos  rt=0,  cos  b=±l. 
These  equations  prove  that  the  resultant  is  equal  to  the  com- 
ponent Y,  and  is  directed  along  the  axis  of  y. 

In  like  manner  it  might  be  shown  that  if  Y  were  equal  to 
zero,  the  resultant  would  be  equal  to  the  component  X,  and 
would  be  directed  along  the  axis  of  a;. 

General  Remarks  on  Forces  situated  in  any  marmer 
in  Space. 

45.  If  three  forces  solicit  a  point,  their  directions  not  being 
confined  to  a  single  plane,  a  theorem  analogous  to  that  of  the 
parallelogram  offerees  will  still  serve  to  determine  their  re- 
sultant. Thus,  let  any  three  forces  P,  P',  and  P"  be  applied 
at  the  point  A  {Fig.  27),  and  let  their  intensities  be  repre- 
sented by  the  lines  AB,  AC,  and  AD.  If  a  parallelopiped  be 
constructed  upon  these  three  lines,  the  diagonal  AE,  of  the 
base  of  this  parallelopiped,  will  evidently  represent  the  re- 
sultant of  the  forces  AB  and  AC  ;  and  by  substituting  the 
force  AE  for  its  two  components,  the  resultant  sought  will 
be  that  of  the  forces  AE  and  AD  ;  it  will  therefore  be  termi- 
nated at  the  extremity  F  of  the  line  EF  drawn  parallel  and 
equal  to  the  line  AD  ;  hence  it  will  be  the  diagonal  of  the 
parallelopiped  DE. 

46.  If  the  three  forces  are  rectangular,  the  angle  ABE  will 
be  a  right  angle,  and  hence  we  obtain 

AE==AB2-|-BE2; 


FORCES    APPLIED   TO    A    POINT.  25 

but  the  triang^le  AEF  being  also  right-angled,  we  have 

AF=»=AE2-fEF2. 
And  by  substituting  for  AE*  its  value  given  above,  we  deduce 

AF='=AB»4-BE='-fEF2. 
Or  by  replacing  BE  and  EF  by  their  equals  AC  and  AD,  we 
finally  obtain 

AF=^(AB«  4-AC^  +AD2), 
or, 

the  resultant  of  the  three  forces  being  denoted  by  R. 

47.  It  has  been  shown  that  any  number  of  forces  lying  in 
the  same  plane  may  always  be  referred  to  two  rectangular 
axes  :  in  like  manner,  we  may  refer  to  three  rectangular  axes 
those  forces  which  are  situated  in  different  planes.  Thus, 
having  assumed  three  co-ordinate  axes  passing  through  any 
point  O  {Fig-.  28),  we  draw  through  A,  the  point  of  applica- 
tion of  a  force  P,  the  three  rectangular  axes  Ax,  Ay,  and  Az, 
parallel  respectively  to  the  axes  of  co-ordinates  ;  and  denoting 
by  X,  /3,  y  the  angles  formed  by  AD,  the  direction  of  the  force 
P,  with  the  three  lines  A.v,  Ay,  Az,  the  direction  of  the  force 
will  be  determined  when  these  angles  become  known. 

48.  The  values  of  these  angles  may  also  be  employed  to 
determine  the  components  of  the  force  P,  which  act  in  direc- 
tions parallel  to  the  three  co-ordinate  axes.  For,  DC  beins: 
perpendicular  to  the  plane  i/Ax,  the  angle  DCA  will  be  a  right 
angle,  and  the  triangle  ADC,  having  the  angle  D=y,  will  give 

DC=AD  cos  y (6). 

In  like  manner,  the  components  parallel  to  Ax  and  Ay  will 
be  expressed  by 

AB=AD  cos  u,  BC=AD  cos  |3 (7). 

^nd  replacing  the  line  AD  by  the  force  P  which  it  repre- 
sents, we  obtain  for  the  three  rectangular  components  of  P, 

P  cos  «,  P  cos  /3,  P  cos  y. 

49.  It  is  important  to  observe  that  the  values  of  two  of  the 
angles  u,  jâ,  and  y  will  serve  to  determine  that  of  the  third. 
For,  since  the  square  of  the  diagonal  AD  is  equal  to  the  sum 
of  the  squares  of  the  three  edges,  we  have 

AB='+BC2-1-DC2=AD2  ; 
and  substituting  in  this  equation  the  values  obtained  from  the 

o 
O 


26 


STATICS. 


equations  (6)  and  (7),  suppressing  the  common  factor  AD*, 
there  will  remain 

C0S2<«4-C0S2/3-}-C0S=y  =  l  j 

whence, 

cos  y=±y/(l  — C0S  =  «— C0S'»/3) (8). 

And  since  a  similar  value  may  be  found  for  each  of  the  other 
cosines,  it  follows  that  the  angle  formed  by  the  direction  of  a 
force  with  either  of  the  axes  will  become  known,  when  the 
angles  formed  with  the  other  two  axes  have  been  previously 
determined. 

50.  The  radical  in  equation  (8)  being  affected  with  the 
double  sign,  the  cosine  of  y  may  be  either  positive  or  nega- 
tive. The  first  value  will  obtain  when  the  angle  is  acute, 
and  the  second  when  it  is  obtuse. 

But  the  angle  y  will  be  acute  or  obtuse  according  to  the 
position  of  the  force  P  ;  in  the  first  case,  the  force  falls  above 
the  plane  xky,  and  the  co-ordinates  z  of  the  points  in  the  line 
representing  the  force,  will  therefore  be  positive  ;  in  the 
second,  it  falls  below  xAy,  and  the  co-ordinates  z  will  then 
be  negative. 

The  same  observations  may  be  extended  to  the  angles  » 
and  /3  considered  with  reference  to  the  axes  o(  x  and  y  ;  so  that 
in  general  the  cosines  will  be  affected  with  the  same  signs  as 
the  co-ordinates  x,  y,  z,  reckoned  from  A. 

51.  The  signs  of  the  cosines  may  also  be  determined  by  a 
rule  which  is  founded  on  Art.  10.  Thus,  if  Ax  {Fig.  29) 
represent  the  line  of  direction  of  a  component,  this  compo- 
nent will  be  positive  when  it  acts  in  the  direction  from  A 
towards  x,  but  negative  if  it  acts  from  A  towards  x'.  The 
tendency  of  the  force  in  the  first  case  will  be  to  remove  the 
point  A  from  the  origin  O,  but  in  the  second  to  cause  its  ap- 
proach. Hence,  we  derive  the  following  rule  :  A  compoiiem 
is  positive  when  it  tends  to  increase  the  co-ordinate  of  th( 
point  of  application,  and  negative  when  it  tends  to  diniinisl 
this  co-ordinate. 


FORCES    APPLIED   TO   A    POINT.  27 


Of  Forces  situated  in  Space,  and  applied  to  a  Point. 

52.  Let  P,  P',  P",  &c.  represent  different  forces  which  so- 
licit a  point  A,  and  let  there  be  drawn  through  this  point  the 
three  rectangular  axes  Kx,  Ay,  Az  ;  represent  by 

«,  /3,  y,  the  angles  formed  by  the  force  P  with  the  axes  of  co- 
ordinates, 
«',  |3',  y',  the  angles  formed  by  P'  with  the  same  axes, 
«*',/3",y",  the  angles  formed  by  P"  with  the  same  axes, 

&,c.  &c.  &c. 

By  resolving  these  forces  into  components  acting  along  the 
three  axes,  we  shall  obtain  (Art.  48) 

P  cos  «,  P  cos  /3,  P  cos  y,  compoueuts  of  P, 
P'  cos  et'j  P'  cos  Q',  P'  COS  y',  Components  of  P', 
P"cos«", P" cos ô",P", cosy",  Components  of  P". 
If  we  defer,  as  in  Art.  40,  the  determination  of  the  signs  of 
the  cosines  of  these  angles  until  the  formulas  are  applied  to 
a  particular  example,  and  denote  by  X,  Y,  and  Z  the  com- 
ponents of  the  resultant,  directed  along  the  three  axes,  we 
shall  have 

P  cos  «+F  cos  a'+P"  cos  «"-F&c.=X (9), 

P  cos  /3  +  P'  cos  /3'  +  P"  COS  /3"  +  &c.  =  Y (10), 

P  COS  y+P'  cos  y'  +  P"  COS  y"+&C.  =  Z (11), 

53.  But  X,  Y,  and  Z  being  the  projections  AB,  BC,  and 
CD  of  the  right  line  AD,  which  represents  the  resultant  R 
{Fig.  28),  we  shall  obtain  (by  Art.  46) 

AB^ -f  BC^ +CD2  =AD% 

and  consequently, 

X2-fY2+Z='=R^ 
The  intensity  of  the  resultant  will  thus  be  determined,  being 
expressed  by  the  equation 

R=^(X2+Y='-f-Z») (12). 

Again,  if  we  call  a,  6,  and  c  the  angles  formed  by  the  result- 
ant with  the  co-ordinate  axes,  the  components  of  R  directed 
along  the  axes  will  be 

R  cos  a,  R  cos  b,  R  cos  c  ; 


28  STATICS. 

and  since  these  components  have  been  represented  by  the 
quantities  X,  Y,  and  Z,  we  shall  have 

X=R  cos  a,    Y=R  cos  6,    Z=Rcosc; 
whence, 

X  ,     Y  Z 

cos  «— p">    COS  0=-jîj    cosc  =  :5- (13). 

If  the  forces  P,  P',  P",  &.C.,  and  the  angles  a,  js,  y,  «',  (3',  y',  dec 
are  known,  the  values  of  X,  Y,  and  Z  will  result  from  the 
equations  (9),  (10),  and  (11).  These  values  being  substituted 
in  formula  (12),  the  intensity  of  the  resultant  will  be  deter- 
mined, and  its  position  will  become  known  from  the  equa- 
tions (13). 

54.  If  an  equilibrium  subsists,  the  resultant  becomes  equal 
to  zero,  and  the  equation  (12)  then  assumes  the  form 

X=+Y3-fZ2=0. 
And  since  this  equation  cannot  be  true  unless  the  terms  are 
separately  equal  to  zero,  we  have 

X=0,  Y=0,  Z  =  0. 
These  values  reduce  the  equations  (9),  (10),  (11)  to 

P  cos  «-f-P'  cos  a'  +  P"  cos  <«"-}-&.C.  =  0  i 

P  cos  i3  +  P'  cos  /3'+P'  COS  /3"-l-&c.=0  >  (14). 

P  COS  y+P'  COS  y'-fP"  cos  y"-|-&c.=0  7 
Such  are  the  conditions  of  equilibrium  of  a  system  of  forces 
situated  in  any  manner  in  space,  and  applied  to  a  point. 

55.  If  we  determine  the  resultant  of  all  the  forces  in  the 
system  except  one,  the  remaining  force  will  be  found  equal 
and  directly  opposed  to  this  resultant.  For,  let  R'  represent 
the  resultant  of  all  the  forces  except  P  ;  X',  Y',  and  Z  its 
three  components,  and  a,  b\  and  c  the  angles  which  its  direc- 
tion forms  with  the  co-ordinate  axes  ;  we  shall  have 

X'=P'  cos  «'+P"  cos  «"-f  P"  cos  «'"-f&c, 
Y'=P'  cos  /s'-f  P"  cos  i3"4-P"  cos  /3  "+&C., 
Z'=P'  cos  y'-f  P"  cos y"-f  P" cos  y"+&C.  , 
and  by  means  of  these  values  the  equations  (14)  may  be  re- 
duced to 

P  cos  «-f  X'=0, 
Pcos/3-fY'=0, 
Pcosy-fZ'=0; 


FORCES   APPLIED   TO   A   POINT.  29 

and  eliminating  X',  Y',  Z',  by  the  equations 

X'=R'  cos  a,  Y'=R'  cos  6',  Z  =R'  cos  c', 
there  results 

P  cos  «=— R'  cos  d  \ 

P  cos  /3=  — R'  cos  6'  > (15). 

P  cos  y  =  — R'  cos  c   j 
Taking  the  sum  of  the  squares  of  these  three  equations,  we 
obtain 

P*(cos2«-f  cos»/3-f  cos'y)=R'2(cos2a'4.cos26'4-cos2c')  ; 
and  since  the  second  factor  in  each  member  is  equal  to  unity,    /  ^ 
this  equation  reduces  to 

P3=R2,  or  P=R'. 
The  force  P  is  regarded  as  essentially  positive,  its  position 
being  determined  by  the  rule  explained  in  Art.  35,  &c 

If  the  value  of  P  be  substituted  in  equations  (15),  the  factor 
R'  being  suppressed,  those  equations  will  become 

cos  «= — cos  a (16), 

cos /3=— cos  h' (17), 

cos  y= — cos  c (18). 

The  relation  between  the  values  of  cos  «  and  cos  a  indicates 
that  d  and  «  are  supplements  of  each  other.  For,  if  cos  a' 
1)6  represented  by  AC  [J^ig.  30),  cos  a.  will  be  represented  by 
AC'=AC  ;  whence  a'=DAC,  and  «=DAO. 

But  these  two  angles  are  supplements  of  each  other  ;  for, 
AC  being  equal  to  AC,  gives  the  angle  DAC=D'AC';  whence, 
by  substituting  this  value  in  the  equation 

DAC+DAC'=2  right  angles, 
we  get 

D  AC4-DAC=2  right  angles, 
or  the  angles  a  and  «  are  supplements  of  each  other. 

In  the  same  manner  may  it  be  proved  by  the  equations 
(17)  and  (18),  that  the  angles  h'  and  0  are  supplements  of 
each  other,  as  also  are  the  angles  c  and  y. 

It  results  from  what  precedes  that  the  forces  P  and  R'  are 
directly  opposed  ;  for,  if  R  be  supposed  situated  above  the 
plane  of  a:,  y,  having  the  co-ordinates  x  and  y  both  positive, 


30  STATICS. 

P  will  be  situated  below  this  plane,  and  will  have  the  co-ordi- 
nates X  and  y  both  negative. 

56.  After  reducing  all  the  forces  to  three  rectangular  com- 
ponents X,  Y,  Z,  it  was  shown  that  the  resultant  R  would 
be  represented  by  the  diagonal  of  a  parallelopiped,  whose 
contiguous  edges  were  respectively  equal  to  X,  Y,  and  Z 
{Fig.  27).  The  equation  of  this  resultant,  which  is  repre- 
sented by  AF,  will  therefore  be  that  of  a  right  line  passing 
through  A,  the  origin  of  co-ordinates,  and  through  the  point 
F,  whose  co-ordinates  are  equal  to  X,  Y,  and  Z. 

57.  The  case  may  be  rendered  yet  more  general  by  sup- 
posing that  the  point  of  application  of  the  forces  has  the 
three  co-ordinates  x,  y\  and  z  ;  the  co-ordinates  of  the  point 
F  will  then  become  {Fig.  31) 

y-f  X,  y'-FY,  z'-^Z. 
And  the  equations  of  the  resultant,  being  that  of  a  right  line 
in  space,  will  be  of  the  form 

z=ax-\-h,    z=ay-\-b' (19)  : 

substituting  in  these  equations  the  co-ordinates  of  the  point 
F;  in  place  of  the  quantities  x,  y,  and  z,  we  find 

z'-{-Z=ax'-\-aX+b,   z'-\-Z  =  ay'+àY+b' (20); 

but  the  co-ordinates  of  the  point  A  should  also  satisfy  the 
equations  (19),  and  therefore  we  obtain 

z'=ax'+b,    z'=ay-\-b' (21). 

Subtracting  these  last  from  equations  (20),  we  have 

Z^aX,  Z=a'Y; 
whence, 

Z      ,   Z 
a=X>  «=Y- 

Again,  by  eliminating  è  and  h'  between  the  equations  (19) 
and  (21),  we  find 

z—z'=a{x—x'y,  z — z'=a'{y—y'): 
and  by  substituting  the  values  of  a  and  a'  previously  ob- 
tained, the  equations  of  the  resultant  finally  become 

z-z~{x-xl  z-z'=^{y-y']. 


P 


EQUILIBRIUM  OP  A  POINT  UPON  A  CURVED  SURFACE.     3^1 

Of  the  Conditions  of  Equilibrium  of  a  Point  acted  upon 
by  several  Forces,  and  subjected  to  the  Condition  of  re- 
maining upon  a  Given  Surface. 

58.  The  material  point  to  which  the  forces  P,  F,  P",  &c. 
were  applied,  has  been  supposed  hitherto  to  submit  freely  to 
the  action  which  those  forces  exert  ;  but  if,  on  the  contrary, 
the  point  were  required  to  remain  constantly  on  a  given  sur- 
face, the  equations  (14)  would  no  longer  be  applicable,  and 
the  condition  of  the  resultant  being  equal  to  zero,  which  was 
then  necessary,  would,  under  this  supposition,  be  replaced  by 
the  condition  that  the  resultant  must  be  normal  to  the  given 
surface.  For,  if  the  direction  of  the  resultant  be  oblique 
to  the  surface,  it  can  be  decomposed  into  two  forces,  of  which 
one  shall  coincide  with  the  direction  of  the  tangent,  and  the 
other  with  the  normal  :  the  first  would  cause  the  material 
point  to  slide  along  the  surface,  while  the  second  would  be 
overcome  by  the  reaction  of  the  surface.  Hence,  it  follows 
that  the  resultant  of  all  the  forces  must  act  on  the  point  in 
the  direction  of  the  normal  to  the  surface,  and  since  the  re- 
sultant is  destroyed  by  the  resistance  of  the  surface,  we  may 
regard  this  resistance  as  a  force  directly  opposed  to  the  nor- 
mal force,  and  denote  its  intensity  by  a  quantity  N. 

If  the  intensity  of  the  force  N  and  the  angles  6, 6',  6%  which 
it  forms  with  the  co-ordinate  axes,  were  known,  it,  would  be 
sufficient  to  add  to  the  equations  of  equilibrium  the  compo- 
nents N  cos  5,  N  cos  6\  N  cos  6"  of  the  force  N  ;  we  should  thus 
obtain  the  equations  of  equilibrium 

N  cos  6-\-V  cos  «s-f  P'  cos  x-j-V"  cos  «"-|-&c.— 0, 

N  cos  6'  +  F  cos  /3-f-P'  cos  B'  +  V  COS  (3"-f  &c.  =  0, 
N  COS  Ô'  +  P  COS  y  +  P'  COS  y+P'coS  y  "-{-&C.  =  0. 

59.  These  equations  may  be  simplified  by  representing,  as 
in  Art.  52,  by  X,  Y,  and  Z,  the  sums  of  the  components  par- 
allel to  the  three  axes  ;  the  equations  will  thus  become 

N  cos  ô-|-X=0,   N  cos  o'-f  Y=0,    N  cos  â'-f-Z=0 (22). 

60.  To  determine  the  values  of  the  unknown  quantities  cos 
ê,  cos  6',  COS  ê",  and  N,  we  will  suppose  L=0  to  be  the  equation 
of  the  given  surface,  and  x,  y\  and  z  the  co-ordinates  of  the 


32  STATICS. 

material  point  to  which  the  forces  are  apphed,  and  which  by 
hypothesis  is  required  to  remain  on  this  surface.  The  nor- 
mal being  a  right  hne  passing  through  the  point  whose  co- 
ordinates are  x\  y',  z',  its  equations  will  be  of  the  form 

x~x=a{z—z\  y—y  =  h{z—z') (23). 

The  differences  x — x\  y — y\  z — z,  which  enter  into  these 
equations,  represent  the  projections  of  the  right  line  on  the 
axes  of  co-ordinates.  To  determine  the  relations  existing 
between  these  projections  and  the  angles  <!,  6\  6",  let  MN  {Fig. 
32)  represent  the  right  line  in  space  referred  to  the  co-ordi- 
nate axes  whose  origin  is  at  the  point  O,  and  denote  by  x,  y, 
z,  x\  y',  z',  the  co-ordinates  of  the  points  N  and  M  :  if  a  plane 
DF  be  passed  through  the  co-ordinates  MD—z  and  BD=y', 
and  a  second  plane  EG  through  NE=2;  and  EC =y,  these 
two  planes  will  be  parallel  to  that  of  y,  z,  and  the  distance 
between  them  will  be  measured  by  the  part  'QC—x—x  inter- 
cepted on  the  axis  of  x  :  but  since  every  parallel  to  this  axis 
is  likewise  perpendicular  to  the  two  planes,  it  follows  that  by 
drawing  through  the  point  M,  the  extremity  of  the  co-ordi- 
nate z',  the  parallel  MP  to  the  axis  of  x,  this  parallel  will  be 
perpendicular  to  the  plane  EG,  and  will  intersect  it  at  a  dis- 
tance MP=ar — x'. 

But,  by  connecting  the  point  P  with  N,  the  point  at  which 
the  right  line  MN  intersects  the  plane  EG,  a  triangle  will  be 
formed  right-angled  at  P,  since  MP  is  perpendicular  to  the 
plane  EG.     Hence, 

MP=MN  cos  M, 
or, 

x—x'^MN  cos  6  ; 
but  MN  being  a  right  line  passing  through  the  two  points 
whose  co-ordinates  are  x,  y,  z,  x,  y',  z\  its  length  will  be  ex- 
pressed by 

^[{x-x-y  +{y—yy  Hz-zf]. 

Substituting  this  value  in  the  preceding  equation,  we  deduce 

X — x' 

^°^  ^^  Ai.x-xy-\-{y-yYHz-zYY 

In  like  manner,  by  drawing  planes  through  the  co-ordinates 
x\  z\  and  a:,  z,  parallel  to  the  plane  of  ar,  z^  and  through  x.  y\ 


EQUILIBRIUM  OP  A  POINT  UPON  A  CURVED  SURFACE.     33 

and  X,  y,  parallel  to  the  plane  of  x,  y,  we  shall  find  for  cos  6^ 
and  cos  6",  the  similar  expressions 

,  y — y' 

cos  Ô .—  -  ^     '^ 


cos  6"-. 


z — z' 


■  ^[[x-x'Y  +{:y-yy  +{z-  z'Y] 
by  eliminating  the  values  of  x — x\y — y\  by  means  of  equa- 
tions (23),  and  suppressing  the  common  factor  z — z\  we  obtain 


a  ,  h 

cos  6  = ,     „    ,    ,  „    .   ix;  COS  6  —■ 


'] 


COS  6''-- 


^(«=^+6=^+1)'  -^{a^^  ^b-  +  iy  \^ ^24^ 


61.  These  values,  which  serve  to  determine  the  direction 
of  the  normal,  contain  the  quantities  a  and  b,  which  are  yet 
unknown.  The  values  of  these  quantities  will  now  be  de- 
termined. Let  L=0  be  the  equation  of  the  surface  which 
passes  through  the  point  x',  y\  z  ;  if  we  draw  through  this 
point  a  plane  tangent  to  the  surface,  the  equation  of  this 
plane  will  be  of  the  form 

Kx+By+Cz-\-I>=0] 
and  since  it  must  be  satisfied  by  the  co-ordinates  x\  y\  z',  we 
shall  have 

Ax'+By'-^Cz'+B^O. 

Eliminating  D  between  these  two  equations,  the  equation  of 
the  tangent  plane  to  the  surface  becomes 

A{x-x')+B(y~y')-{-C{z-z)=0  ; 
and  dividing  by  C,  it  may  be  put  under  the  form 

^(x~x'n^{y-y')-{-{z-z')=0 (25). 

But  if  the  plane  be  tangent  to  the  surface  whose  equation  is 

dz'         dz' 
L=0,  the  values  of  -j^,  and  j^  deduced  from  that  equation, 

will  be  expressed  as  follows  : 

dz'         A   dz'        B 

rf?=~"C'  dy'^~C ^^^^• 

And  from  the  known  principles  of  analytical  geometry,  when 
a  plane  whose  equation  is  Ax-\-By-\-Cz+D=0  is  pcrpen- 

C 


34  STATICS. 

dicular  to  a  right  line  represented  by  the  equations  x=az-\-», 
y=hz-\-^,  the  following   relations    between    the   constants 

exist  : 

A  B     ^ 

the  equations  (26)  will  therefore  reduce  to 

£:=_.|;=-......(2T). 

62.  The  values  of  these  coefficients  must  now  be  deter- 
mined from  the  equation  of  the  surface.  We  obtain  by  dif- 
ferentiating, 

d\j  ,    ,  dh  ,    ,  dlj  J       ,, 
dx  dy         dz 

whence  we  infer  that 

dL^        dL 

dz=z  —  '^dx — ^dy  ;      . 
rfL         dL    ^ 

dz  dz 

and  by  applying  this  equation  to  the  point  of  tangency,  for 

which  the  co-ordinates  are  x\  y\  z\  we  find 

dL_  rfL_ 

dz'_  _dx'      dz  _     dy' 

dx'  ^'    ~d^'~~~dO 

dz'  d^ 

substituting  these  values  in  the  equations  (27),  they  become 

dL  dL 

dL'  dL 

dz  dz 

Replacing  a  and  b  in  equations  (24),  by  their  values  found 
above,  we  obtain,  after  reduction, 

dL 


cos  êz=  ± 


dx' 


^/]&-m'^m 


cos  6'=  ± 


dy' 


^liÈr-ar^m'ï 


EQUILIBRIUM  OF  A  POINT  UPON  A  CURVED  SURFACE.     35 

dh 

COS  ^  =  ± 


The  double  sign  is  here  prefixed  to  the  values  of  cos  ^,  cos  ô', 
COS  0",  for  the  purpose  of  indicating  that  the  resistance  op- 
posed by  the  surface  may  be  exerted  either  in  the  direction  of 
the  normal  or  along  its  prolongation,  according  as  the  body  is 
placed  on  the  concave  or  convex  side  of  the  surface.  The 
form  of  these  equations  being  inconvenient  for  the  purposes 
of  calculation,  they  may  be  simplified  by  making 

^  =V (28); 


s/\m^m-{W\ 


which  reduces  them  to 

cosfl=V— ^,    costf=V^— ,    cosfl  =¥-—,; 
ax  ay'  dz 

substituting  these  values  of  the  cosines  in  equations  (22),  we 

obtain 

NV^+X=0,  NV'^.+Y^O,  NV^4.Z=0 (29). 

dx  dy  dz 

63.  The  value  of  IN  remains  to  be  determined.  If  we 
transpose  X,  Y,  and  Z  in  the  equations  (29),  and  take  the 
sum  of  the  squares  of  the  three  equations,  we  shall  obtain 

and  reducing  by  means  of  equation  (28)  there  results 

N2=X2+Y2+Z2, 

whence, 

N=^(X='+Y-+Z=') (30). 

This  value  of  N  is  precisely  the  same  as  that  of  the  resultant 
of  the  entire  system  ;  but  its  components  should  be  aftected 
with  signs  contrary  to  those  of  the  components  of  the  result- 
ant, since  its  action  is  exerted  in  an  opposite  direction. 
Thus,  having  determined  the  resultant  of  all  the  forces  P,  P', 
P",  &c.,  the  reaction  of  the  surface  will  be  equal  to  this  re- 
sultant, but  will  be  exerted  in  an  opposite  direction. 

C2 


36  STATICS. 

64.  If  the  direction  of  the  normal  force  be  parallel  to  the 
axis  of  2r,  we  shall  have 

<j=90=,  ^'=90°,  6"=0,  or  o"=180°  ; 
whence 

cos  0=0,  cos  ô'=0,  cos  $"=  ±  1  : 
and  the  equations  (22)  will  therefore  reduce  to 

X=0,  Y=0,  N±Z=0; 
which  prove  that  the  components  in  the  direction  of  the  tan- 
gent plane  destroy  each  other,  and   that  the  reaction  of  the 
surface  in  the  direction  of  the  normal  is  equal  to  the  sum  of 
the  components  directed  along  the  axis  of  z. 

65.  The  nature  of  the  problem  may  also  be  such  that 
having  given  the  forces  P,  F,  P",  &c.  and  the  equation  of  the 
surface  upon  which  the  material  point  should  rest,  it  might 
be  required  to  determine  x',  y\  and  z^  the  co-ordinates  of  the 
point  at  which  the  forces  should  be  applied  in  order  that  the 
material  point  should  be  sustained  in  equilibrio. 

To  resolve  this  problem,  we  first  eliminate  the  quantity  N, 
by  combining  the  equations  (29)  ;  the  factor  V  will  likewise 
disappear,  and  we  shall  then  have 

d-^         dz      dy        dz  ' 
these  equations,  in  conjunction  with  that  of  the  surface,  will 
serve  to  determine  the  co-ordinates  x',  y\  and  z  of  the  point 
of  application. 

Of  the  Conditions  of  EquiUhrium  of  a  Point  acted  on  by 
several  Forces,  and  subjected  to  the  Condition  of  remain- 
ing constantly  on  two  Curved  Surfaces,  or  on  a  Curve  of 
Double  Curvature. 

66.  If  a  material  point  be  retained  on  two  curved  surfaces, 
it  cannot  remain  in  equilibrio  unless  the  force  which  solicits 
it  can  be  decomposed  into  two  components  which  shall  He 
respectively  normal  to  the  given  surfaces  ;  for,  if  one  of  these 
components  had  a  different  direction,  it  might  be  decomposed 
into  two  forces,  of  which  the  first  normal  to  one  of  the  sur- 
faces, should  be  destroyed  by  the  reaction  of  the  surface,  and 


and 


EQUILIBRIUM  OF  A  POINT  UPON  TWO  SURFACES.  37" 

the  second  tangent  to  the  same  surface,  would  move  the  body- 
along  the  surface. 

Let  N  and  M  represent  the  reactions  of  the  two  surfaces, 
and  6,  e',  e",  «,  „',  «s"  the  angles  formed  by  their  normals  with 
three  rectangular  axes  drawn  through  the  point  to  which  the 
forces  are  applied  :  by  adoptmg  the  same  course  of  reasoning 
as  in  Art.  59,  we  shall  obtain 

N  cos  ê  +M  cos  »»  +X=0  ^ 

N  cos  6'  +M  cos  „'  +Y=0  V (31> 

N  cos  o'+M  cos  V'+Z=0  ) 
The  equations  of  the  surfaces  L=0  and  K=0  being  differ- 
entiated, make  known,  as  in  Art.  62,  the  values  of  the  quan- 
tities cos  ê,  cos  s',  cos  6",  cos  a,  cos  «',  cos  >)",  and  by  adopting 
abbreviations  similar  to  those  of  Art.  62,  making 

± I _=u, 

we  shall  find 

„c?L  ^-,dK 

cos  6=\-T—, ,     COS  t}=\J--—, , 

ax  ax 

,,    T7-c?L  ,    -r^dK. 

cos^=V— ^.   cos;î=IJ--;, 
di/  dy 

^^dL  „     T,dK 

cos  6  =  V-— , ,  COS  !,'=U— -  : 
dz  dz 

which  values,  being  substituted  in  the  equations(31),  give 

dx  dx 

NV^^+MU^-fY=0  y (32). 

dy  dy' 

NV^-fMU^+Z=0 
dz  dz 

From  these  three  equations  the  unknown  quantities  M  and 
N  may  be  readily  eliminated  ;  and  since  U  and  Venter  into 
.them  in  the  same  manner  as  M  and  N,  ihey  will  also  disap- 


38 


STATICS. 


pear  in  the  elimination  :  or,  to  simplify  the  case,  we  may 
regard  MU  and  NV  as  the  unknown  quantities,  which,  being 
eliminated  between  the  three  preceding  equations,  will  give 
an  equation  of  condition  including  one  or  more  of  the  three 
variables.  This  resulting  equation  being  combined  with 
those  of  the  surfaces,  viz.  L=0,  K=0,  will  determine  the  co- 
ordinates .X-',  y',  z\  of  the  point  sought. 

It  may  be  proper  to  remark  that  the  radicals,  which  would 
serve  to  complicate  the  expressions,  disappear  at  the  same 
time  as  the  quantities  U  and  V. 

67.  When  the  point  is  subjected  to  the  condition  of  re- 
maining on  a  curve  of  double  curvature,  such  curve  may  be 
regarded  as  being  formed  by  the  intersection  of  two  curved 
surfaces.  The  equations  of  these  surfaces  being  represented 
as  above  by  L=0  and  K=0,  the  co-ordinates  of  the  points  in 
which  they  intersect  will  necessarily  appertain  to  both  sur- 
faces, and  the  quantities  x\  y\  and  z'  may  therefore  be  re- 
garded as  having  the  same  values  in  each  of  these  equations  ; 
but  we  have  also  the  equation  of  condition  referred  to  in 
Art.  66  ;  thus  by  eliminating  the  values  of  two  of  the  co- 
ordinates, the  third  will  be  expressed  in  functions  of  known 
quantities  :  denoting  by  A,  B,  and  C  the  values  of  the  func- 
tions corresponding  to  each  of  the  co-ordinates  x\  y\  and  z\ 
we  shall  have 

x'—K,  y'=B,   z—G. 

68.  It  may  occur  that  the  equation  resulting  from  the  elim- 
ination of  M  and  N  will  not  contain  either  of  the  variables. 
This  case  presents  itself  when  the  surfaces  become  planes  ; 
their  equations  L=0  and  K=0  may  then  be  put  under  the 
form  Aa:+By+C;::+D=0,  and  the  differential  coefficients  are 
then  constant.  Under  such  circumstances  the  values  of  the 
intensities  M  and  N  determined  by  the  equations  (32)  become 
independent  of  the  co-ordinates  x',  y\  and  z  ;  and  since  these 
co-ordinates  still  apply  to  any  points  common  to  the  two  planes, 
it  follows  that  the  conditions  of  equilibrium  will  be  fulfilled, 
if  the  forces  be  applied  to  any  point  whatever  in  the  common 
intersection  of  the  two  planes.  A  similar  remark  is  appli- 
cable to  Art.  65. 


PARALLEL    FORCES.  39 


Of  Parallel  Forces. 

69.  The  forces  which  have  been  considered  in  the  pre- 
ceding paragraphs  were  supposed  to  have  a  common  point 
of  apphcation  ;  but  if  they  were  appHed  to  different  points 
of  a  body  or  system  of  bodies,  the  points  being  retained  at 
fixed  distances  by  means  of  their  connexion  with  the  inter- 
mediate points,  we  might  regard  the  forces  as  having  their 
points  of  apphcation  united  by  means  of  inflexible  right 
hnes. 

70.  Let  there  be  two  parallel  forces  P  and  Q,  applied  to  the 
extremities  of  a  right  line  AB  {Fig-  34),  which  intersects  at 
right  angles  their  common  direction.  It  has  been  proved 
(Art.  22)  that  the  intensity  of  the  resultant  of  these  forces  will 
be  equal  to  the  sum  of  the  intensities  of  the  two  components, 
and  that  its  point  of  application  O  will  divide  the  line  AB  in 
the  inverse  ratio  of  the  two  forces.  This  proposition  may  be 
demonstrated  in  another  manner,  provided  we  admit  that  of 
the  parallelogram  of  forces,  which  is  susceptible  of  direct 
proof 

Let  the  two  parallel  forces  be  represented  by  the  right  lines 
AP  and  BQ,  proportional  to  their  intensities  {Fig.  33)  ;  we 
can  add  to  the  system,  without  changing  the  value  of  the 
resultant,  the  two  equal  and  opposite  forces  AM  and  BN, 
and  the  four  forces  AP,  AM,  BQ.,  and  BN  may  then  he  re- 
placed by  the  two  AD  and  BI,  the  diagonals  of  the  rectangles 
MP  and  NO..  But  since  these  diagonals  intersect  at  the 
point  C,  the  forces  AD  and  BI  may  be  conceived  to  be  applied 
at  that  point,  and  will  be  represented  by  CE=AD  and 
CF=BI.  If  the  forces  CE  and  CF  be  then  decomposed  into 
rectangular  components,  by  constructing  the  rectangles  GL 
and  HK,  having  their  sides  respectively  equal  and  parallel  to 
those  of  the  rectangles  MP  and  NQ,,  we  shall  replace  CE 
and  CF  by  the  four  forces  CL,  CK,  CG,  and  CH.  But  the 
last  two  are  equal,  being  equivalent  to  the  forces  AM  and 
BN;  which  by  hypothesis  are  equal,  and  being  directly  op- 
posed, they  must  mutually  destroy  each  other  ;  there  will 
tlxerefore  remain  at  the  point  C,  the  two  forces  CL  and  CK 


40  STATICS. 

equal  respectively  to  P  and  Q,,  and  having  the  common  direc- 
tion of  the  hne  CO.  The  resultant  of  these  two  forces  must 
evidently  be  equal  to  their  sum  ;  and  if  ii  be  denoted  by  R, 
we  shall  have  the  relation 

R=P4.Gl: 

but  since  the  resultant  may  be  applied  at  any  point  in  its  line 
of  direction,  we  will  consider  it  as  acting  at  0,  the  point  in 
which  it  intersects  the  line  AB  ;  the  position  of  this  point 
may  be  determined  thus  :  the  two  similar  triangles  CAO, 
CEL  give  the  proportion 

CO  :  AO  :  :  CL  :  EL, 
and  the  triangles  COB,  CKF  give 

BO  :  CO  :  :  KF  :  CK  ; 
whence,  by  multiplication,  suppressing  the  common  factor 
CO,  we  have 

BO:  AO::  CLxKF:ELxCK. 

But  KF  and  EL,  being  equal  to  BN  and  AM,  which  by  hy- 
pothesiG  are  equal  to  each  other,  the  proportion  reduces  to 

BO  :  AO  :  :  CL  :  CK  : 
and  since  CL  and  CK  are  equivalent  to  the  lines  AP  and  BQ, 
which  represent  the  intensities  of  the  given  forces,  the  pro- 
portion may  be  written 

BO  :  AO  :  :  P  :  a (33). 

Hence  we  conclude  that  the  point  of  application  O  of  the  two 
parallel  forces  P  and  Q.  divides  the  line  AB  into  two  parts, 
reciprocally  proportional  to  the  intensities  of  those  forces. 
71.  From  the  above  proportion  we  obtain  {Pig.  34) 

BO+AO  :  AO  ::  P  +  Q  :  Q, 

or, 

AB  :  AO  :  :  R  :  a (34). 

And  from  the  equations  (33)  and  (34)  combined,  we  find 

P  :  a  :  R  :  :  BO  :  AO  :  AB  ; 
from  which  we  derive   the  following  rule  :    T/ie  jyarts  AO, 
BO,  avd  AB  compj'ised  between  any  two  of  the  forces  P,  Q, 
and  R,  will  he  co'nstanily  'proportional  to  the  tJii)'d  force.   The 
term  R,  for  example,  in  the  above  proportion;  corresponds:  to 


PARALLEL   FORCES.  41 

the   portion  AB,  which  is  included  between  the  forces  P 
and  Q,. 

72.  If  from  the  known  values  of  P,  Q,  and  AO,  it  were 
required  to  determine  that  of  BO,  the  proportion  would  give 

a  :  P  :  :  AO  :  BO  ; 
whence, 

BO=P^_^. 

a 

73.  Reciprocally,  if  there  were  given  the  force  R  applied  at 
O,  and  we  wished  to  resolve  it  into  two  parallel  components 
whose  points  of  application  should  be  A  and  B  ;  by  denoting 
the  unknown  components  by  P  and  Q,  the  value  of  the  first 
would  result  from  the  proportion 

AB  :  BO  :  :  R  :  P  ; 
and  that  of  the  second  would  in  like  manner  be  obtained  by 
means  of  the  proportion 

AB  :  AO  :  :  R  :  CI. 

From  these  two  proportions  we  deduce 
p_RxBO  RxAO 

^~~AB~'     ^        AB~' 

In  the  preceding  demonstration,  the  forces  P  and  Q.  have 
been  supposed  perpendicular  to  the  line  AB  ;  but  if  they  were 
oblique  to  the  direction  of  this  line,  we  might  draw  through 
O,  the  point  of  application  of  the  resultant  {Fig.  35),  the  right 
line  CD,  perpendicular  to  the  direction  of  the  given  forces, 
and  the  force  P  applied  at  A  would  have  the  same  effect  as 
though  it  were  applied  at  the  point  C.  In  like  manner,  the 
point  of  application  of  the  force  Q,  may  be  transferred  from 
B  to  D  :  and  since  we  have  the  proportion 

P  :  a  :  :  OD  :  OC, 
we  shall  likewise  obtain  from  the  similarity  of  the  triangles 
OBD,  AOC, 

P  :  a  :  :  BO  :  AO. 

74.  When  the  forces  P  and  Q.  act  in  opposite  directions,  the 
resultant  is  equal  to  the  difference  of  these  forces.  For,  let 
S  {Pig.  36)  be  the  resultant  of  the  forces  P  and  R,  which  are 
supposed  to  act  in  the  same  direction,  we  shall  then  have 


42 


STATICS. 


S=P+R (35); 

and  if  we  replace  S  by  a  force  Q,  equal  in  intensity,  and 
directly  opposed  to  it,  an  equilibrium  will  subsist  between  the 
three  forces  P,  R,  and  Q,  :  we  may  therefore  regard  R  as 
beirig-  equal  and  directly  opposite  to  the  resultant  of  the  forces 
P  and  Q,  and  the  equation  (35)  will  give  for  the  intensity  of 
this  resultant 

R=S-P; 
but  S  and  Q,  being  equal  in  intensity,  we  have,  by  substi- 
tuting the  value  of  S, 

R=Q— P. 
The  point  O  at  which  the  resultant  is  applied,  may  be 
found  by  the  proportion 

AB  :  BO  :  :  R  :  Q, 
whence  we  obtain 

TJ/-V AL  XQ, 

R~' 
or,  replacing  R  by  its  equal  Q— P,  we  have 

QXAB 

From  this  value  of  the  distance  BO,  we  infer  that  the  point 
O  will  be  farther  removed  from  B  in  proportion  to  the  dimi- 
nution of  the  quantity  Q,— P  ;  if  therefore  Q  and  P  become 
equal,  BO  becomes  infinite,  and  R  becomes  equal  to  zero  : 
hence,  if  two  parallel  and  equal  forces  act  in  contrary  direc- 
tions, but  are  not  directly  opposed,  the  equilibrium  cannot  be 
established  except  by  the  application  of  an  infinitely  small 
force  at  a  point  whose  distance  is  infinite  ;  it  is  therefore  im- 
possible in  such  cases  to  find  a  single  finite  force  which  shall 
sustain  in  equilibrio  the  two  forces  P  and  Q,  ;  or,  in  other 
words,  the  two  forces  P  and  Q,  cannot  be  replaced  by  a  single 
resultant.  The  efifect  of  these  forces  will  be  simply  to  turn 
the  line  AB  about  its  middle  point  C. 

75.  These  pairs  of  parallel  and  equal  forces,  acting  in  con- 
trary directions,  but  not  directly  opposed,  are  called  couples. 

76.  The  results  obtained  in  the  preceding  articles  may  be 
applied  to  any  number  of  forces.  Thus,  let  P,  P',  P",  F",  P"', 
{Fiff.  37)  represent  parallel  forces  applied  to  the  points  A,  B, 


PARALLEL   FORCES.  4S 

C,  D,  E,  which  are  connected  together  by  inflexible  right 
lines  ;  the  point  of  appHcation  and  the  intensity  of  the  result- 
ant may  be  readily  found.  For,  the  forces  P  and  P'  being 
compounded,  their  resultant  will  be  applied  at  a  point  M, 
whose  position  may  be  determined  by  the  following  proportion, 

AB:  AM::P4-P':P'5 
whence, 

J^_ABXF 

the  line  MC  being  then  drawn,  we  can  determine  the  point 
of  application  N  of  the  resultant  of  the  forces  P-fP'  applied 
at  M,  and  of  the  force  P"  applied  at  C  ;  for  we  have 

MC:MN::P+P'+P":P"; 
from  which  the  value  of  MN  results, 

mcxp:^ 

P+P'+P" 

By  connecting  the  points  N  and  D,  the  point  of  application 
O,  of  the  four  forces  P,  P',  P",  P  ",  may  be  found  in  a  manner 
precisely  similar,  and  lastly,  by  joining  the  points  O  and  E, 
we  shall  determine  the  point  K  at  which  the  resultant  of  the 
entire  system  must  be  applied. 

77.  When  some  of  the  forces  of  which  the  system  is  com- 
posed exert  their  efforts  in  a  contrary  direction,  we  reduce  the 
components  P,  P',  P",  «fee,  which  are  supposed  to  act  in  the 
same  direction,  to  a  single  resultant  equal  to  their  sum,  and 
likewise  the  components  Q,  Q.',  Q,",  (fee,  which  are  supposed 
to  act  in  a  contrary  direction,  to  a  second  resultant  equivalent 
to  their  sum  ;  then,  having  determined  the  points  of  applica- 
tion K  and  L  {Fig.  38)  of  these  two  resultants,  the  system 
will  be  reduced  to  two  parallel  forces,  the  one  applied  at 
K,  and  equal  to  P+P'+P"  (fee,  the  other  at  L,  and  equal 
Q,+Q,+(^"  «fee.  :  the  resultant  of  these  two  forces  may  then 
be  determined  by  the  method  explained  in  Art.  74. 

78.  If  the  forces  P,  F,  P",  P'",  (fee.  {Fig.  39),  retaining 
their  points  of  application,  and  continuing  parallel,  assume 
the  positions  AQ,  BQ,',  CQ",  DQ,",'  «fee,  the  resultant  will  be 
parallel  to  the  new  directions  of  the  forces,  but  its  intensity 
and  point  of  application  will  remain  unchanged  ;   for,  the 


44  STATICS. 

construction  employed  to  determine  this  resultant,  being- 
dependent  only  on  the  intensities  of  the  forces  and  their 
points  of  application,  the  data  of  the  problem  will  remain  the 
same. 

79.  If,  for  example,  the  forces  P  and  P'  should  assume  the 
positions  represented  by  the  parallels  Ad  and  BCi,'  ;  there 
would  be  given  P,  P',  and  the  line  AB,  to  determine  the  posi- 
tion of  the  point  M  ;  and  this  would  be  determined  from  the 
same  proportion  as  when  the  forces  were  directed  along  the 
lines  AP  and  BP'. 

The  point  through  which  the  resultant  of  a  system  of  par- 
allel forces  constantly  passes,  whatever  may  be  the  direction 
of  those  forces,  is  called  the  centre  of  parallel  forces. 

80.  To  determine  the  co-ordinates  of  the  centre  of  parallel 
forces,  let  P,  P',  P",  &c.  represent  the  intensities  of  the  several 
forces,  and  denote  by  " 

a;,  y,  z^  the  co-ordinates  of  the  point  of  application  M  of 
the  force  P, 

x\  y\  z' those  of  M', 

x",y",z" those  of  M", 

a:,,  yi,  Zi, those  of  the  centre  of  parallel  forces. 

If  we  represent  by  N  {Fig.  40)  the  point  of  application  of 
the  resultant  of  the  two  forces  P  and  P',  we  shall  have 

MM'  :  M'N  :  :  P-f  P'  :  P  ; 
and  by  drawing  the  line  ML'  parallel  to  HH',  the  projection 
of  MM'  on  the  plane  of  x,  y,  the  similar  triangles  ML'M', 
NLM'  will  give 

MM'  :  M'N  :  :  ML'  :  NL  ; 
whence,  by  combining  the  two  proportions, 
ML'  :  NL  :  :  P+F  :  P  ; 
from  which  results  the  equation 

(P+P')NL=PxML': 
adding  to  each  member  the  product  (P+P')LK,  we  have 

(P+P  )(NL+LK^ =P(ML'+LK)+P' X  LK  ; 
and  since 

NL-{-LK=NK, 


PARALLEL    FORCES.  45 

ML'+LK=MH, 
LK=M'H', 

the  preceding  equation  may  be  reduced  to 

(P+F)NK=PxMH4-FxM'H'. 
If  we  denote  by  Q,  the  resultant  of  the  two  forces  P  and  P', 
and  by  Z  the  co-ordinate  of  its  point  of  application,  this 
equation  may  be  written  under  the  form 

QZ=P;r+PV- 
in  Uke  manner,  representing  by  Q,'  the  resultant  of  the  paral- 
lel forces  Q.  and  P",  and  by  Z'  the  co-ordinate  of  the  point  at 
which  it  is  applied,  we  obtain 

a'Z'=az-hPV; 

and  thence,  by  substitution, 

Gi'Z':=Fz+l?'z'+V"z". 

If  the  resultant  of  the  entire  system  be  represented  by  R^ 
and  the  co-ordinate  of  its  point  of  application,  parallel  to  the 
axis  of  ;r,  by  z^,  we  shall  obtain,  by  continuing  the  same  pro- 
cess, the  general  relation 

Rz,=Pz-fPV+P"z"+&c (36). 

81.  The  7nom,ent  of  a  force  with  reference  to  a  'plane 
is  the  jiroduct  of  the  intensity  of  this  force  hy  the  distance 
of  its  point  of  application  from  the  plane.  The  preceding 
equation  therefore  expresses  that  the  moment  of  the  residtant 
of  the  parallel  forces  P,  P',  P",  ^«c,  taken  with  reference  to 
the  plane  of  x,  y,  is  equal  to  the  sum  of  the  moments  of  the 
several  forces  taken  with  reference  to  the  same  plane. 

The  moments  being  taken  with  reference  to  the  other  two 
co-ordinate  planes,  we  have 

Ry.=Py+Py+PV'+(fec (37). 

R2:,=Px-{-PV-f  PV'-|-&c (38). 

82.  When  the  co-ordinates  x,  y,  z,  x,  y\  z\  &c.  of  the  points 
of  application,  and  the  intensities  P,  P',  P',  &c.  of  the  forces, 
are  given,  the  intensity  of  the  resultant  will  become  known, 
being  equal  to  the  algebraic  sum  of  the  several  intensities  ; 
and  the  values  of  the  co-ordinates  .t,,  y,,  and  sr,,  of  the  centre 
of  parallel  forces,  will  be  found  from  the  equations  (36),  (37)i 
and  (38). 


46  STATICS. 

83.  The  forces  are  affected  with  the  positive  or  negative 
sign  according  to  the  directions  in  which  they  act  ;  and  since 
the  signs  of  the  co-ordinates  are  Ukewise  determined  by  their 
positions  with  respect  to  the  origin  of  co-ordinates,  the  mo- 
ments of  the  forces  must  be  regarded  as  positive,  when  the 
forces  and  co-ordinates  have  the  same  sign,  but  negative 
when  the  two  have  contrary  signs. 

84.  If  the  several  points  of  application  M,  M',  M",  &c.  were 
situated  in  the  same  plane  MM"  {Fig-  41),  the  plane  of  x,  y 
might  then  be  assumed  parallel  to  that  in  which  the  forces 
are  applied,  and  the  co-ordinates  z,  z',  z",  &c.,  being  com- 
prised between  two  parallel  planes,  we  should  have 

z=z'=z"=<fcc.  : 
hence,  if  z,  represent  the  co-ordinate  of  the  centre  of  parallel 
forces,  its  value  will  also  be  equal  to  z  ;  for,  its  extremity 
must  be  found  in  the  plane  MM",  being  determined  by  a  con- 
struction similar  to  that  in  Art.  76.  Thus  the  quantity  z 
becomes  a  common  factor  in  the  equation  (36)  which  then 
reduces  to 

R=P4-P'-fP"-i-&c. 

85.  If  the  points  of  application  were  situated  on  the  right 
line  AB  {Fig.  42),  which  we  will  suppose  parallel  to  the  axis 
of  X,  we  should  have  at  the  same  time 

z=z'=z"=&c.,  and  y=y'=y"=éLC.  ; 
the  equations  (36)  and  (37)  would  then  reduce  to 

R=P4-P'+P"-}-&c (39), 

and  there  would  remain  but  the  single  equation 

R:r,=Pa:-f  PV+PV'-f&c (40). 

In  this  case,  we  may  dispense  with  the  consideration  of  the 
three  axes,  it  being  only  necessary  to  estimate  the  co-ordinates 
X,  x\  x'\  &c.  along  the  line  AB,  to  which  the  forces  are 
applied. 

For  example,  if  we  had 

a:=9,     .t'=3,     a;"=— 3,     a:"'= — 4. 
P=-iP,     P"=-|P,     P"'=2P; 
by  substituting  these  values  in  the  equations  (39)  and  (40) 
we  should  deduce 


PARALLEL    FORCES.  47 

R=p_^P_  |P-f  2P=2P, 
Ra:,=9xP-3xiP+3x|P— 4x2P=lx2P; 

whence, 

.T,=l. 

86.  For  the  purpose  of  determining  the  conditions  of  equi- 
hbrium  of  parallel  forces,  we  shall  adopt  as  most  convenient 
that  position  of  the  axes  in  which  one  of  the  co-ordinate 
planes  is  perpendicular  to  the  direction  of  the  forces  :  let  this 
be  the  plane  of  .r,  y.  Having  reduced  all  the  forces  which 
act  in  the  same  direction  to  a  single  resultant  R^  {Fig-  43), 
and  those  which  act  in  a  contrary  direction  to  a  second  re- 
sultant R^^,  an  equilibrium  will  take  place  in  the  system  when 
the  two  resultants  are  equal  and  directly  opposed. 

The  latter  condition  will  be  fulfilled  when  the  distance 
C'C"  is  equal  to  zero,  which  requires  that  the  co-ordinates  x, 
and  y,  of  the  point  C  should  be  respectively  equal  to  x,,  and 
y^,  those  of  the  point  C". 

Hence,  we  obtain 

The  condition  of  equality  between  the  two  resultants  will  be 
satisfied  when  we  have 

R.  =  -R (41); 

and  we  obtain  by  multiplication 

'S.,x,=-\,x, (42), 

R.y.=-R.y. (43). 

If  we  denote  by  P,  P',  P",  &c.  the  components  of  R,,  and 
by  P'",  P%  (fcc.  the  components  of  R,„  the  property  of  the 
moments  will  give  the  two  equations 

R,*-,  =P:r  +  V'x'  ■\-  V"x"  ^  &c., 
R^^x,, = V"'x"'  -f  P'" X''  -f  P^x^  -f-  (fcc.  ; 
and  substituting  these  values  in  equation  (42),  it  reduces  to 
Pa;  +  PV-fP"^"-t-FV'  +  P"'x''+P'':c'-H&c.=0 (44). 

By  the  same  course  of  reasoning,  the  equation  (43)  may  be 
reduced  to 

Py+py +P'y'4-P'Y"+P'>"  +py +&c.=:0 (45). 


48 


STATICS. 


And  finally,  the  values  of  R,  and  R„   being  substituted  in 
equation  (41),  give 

P  +  F  +  P"  +  F"  +  P"+P''  +  <fcc.=:0 (46). 

87.  If  the  equations  (44),  (4.5),  and  (46)  are  satisfied,  the  sys- 
tem of  forces  will  be  in  equilibrio.  The  conditions  expressed 
by  these  equations  may  be  enunciated  as  follows  :  Aji  equi- 
librium will  subsist  in  a  system  of  parallel  forces,  if  the  sum 
of  the  moments  taken  with  reference  to  each  of  two  rectangu- 
lar planes  parallel  to  the  cominon  direction  of  the  forces,  is 
equal  to  zero  ;  the  sum  of  the  forces  being  at  the  same  time 
equal  to  zero. 

88.  An  equilibrium  will  also  take  place  if  the  resultant  of 
the  system  be  supposed  to  pass  through  a  fixed  point,  since 
the  effect  of  this  resultant  will  then  be  destroyed  by  the  re- 
sistance opposed  by  the  fixed  point. 

Of  Forces  situated  iji  the  same  Plane,  and  applied  to  Points 
connected  together  in  an  invariable  manner. 

89.  Let  P,  P',  P",  P'",  &c.  {Fig.  44)  represent  several  forces 
situated  in  the  same  plane,  and  applied  to  the  points  A,  B,  C, 
D,  &c.,  which  are  supposed  to  be  connected  in  an  invariable 
manner.  If  the  system  admits  of  a  single  resultant,  its  po- 
sition and  intensity  may  be  readily  obtained  by  means  of  the 
following  graphic  construction  : — Having  assumed  the  por- 
tions Aa,  B6,  Cc,  and  Dc?  proportional  to  the  intensities  of 
the  respective  forces,  prolong  the  lines  Aa  and  B6  until  they 
intersect  at  the  point  G,  and  apply  the  forces  P  and  P'  at 
this  point.  Construct  the  parallelogram  GG',  having  its 
sides  respectively  equal  to  Aa  and  Bè,  and  its  diagonal  GG' 
will  represent  in  direction  and  intensity  the  resultant  of  the 
two  forces  P  and  P'  ;  again,  by  prolonging  GG'  and  Cc  until 
they  intersect,  and  constructing  the  parallelogram  HH',  whose 
sides  shall  represent  the  forces  GG'  and  Cc,  the  diagonal  HH' 
will  represent  the  resultant  of  these  forces,  and  will  therefore 
be  the  resultant  of  the  three  forces  P,  P',  and  P".  Lastly,  by 
finding  the  intersection  of  HH'  and  Drf,  and  forming  a  thiid 


FORCES    APPLIED   TO    DIFÎ'ERENT    POINTS.  49 

parallelogram,  its  diagonal  11'  will  represent  the  resultant  of 
the  entire  system. 

90.  If  by  this  construction  we  should  find  one  or  more  pairs 
of  parallel  forces,  the  resultant  may  be  determined  by  the 
methods  explained  in  Arts.  (71),  (72), and  (74),  and  its  intensity 
will  be  equal  to  the  sum  or  difference  of  the  forces.  If  the 
system  contain  two  parallel  and  equal  forces,  acting  in  con- 
trary directions,  but  not  directly  opposed,  we  may  combine 
one  of  them  with  the  other  forces,  and  the  construction  of 
Art.  (89)  may  then  be  continued  ;  but  if  the  entire  system 
can  be  reduced  to  two  equal  resultants  acting  in  parallel 
and  contrary  directions,  but  not  directly  opposed,  we  con- 
clude, as  in  Art.  74,  that  a  single  resultant  cannot  be  obtained. 

91.  If  the  construction  should  give  a  resultant  equal  to 
zero,  an  equilibrium  would  subsist  throughout  the  system.     , 

92.  The  preceding  construction  is  equivalent  to  supposing 
the  forces  applied  at  the  point  I,  in  lines  parallel  to  their  primi- 
tive directions,  and  then  compounding  them  into  a  single  result- 
ant. For,  by  considering  the  forces  P,  Q,,  and  S  {Fig.  45), 
the  resultant  DC  of  the  forces  P  and  Q,  being  applied  at  the 
point  D'  in  its  line  of  direction,  may  there  be  decomposed  into 
the  two  components  D'P'  and  D'Q,',  parallel  and  equal  to 
P  and  a. 

93.  To  determine  the  analytical  conditions  of  equilibrium 
in  a  system  offerees  disposed  like  the  preceding,  we  will  first 
consider  the  case  of  three  forces  P,  P',  and  P",  applied  to 
points  which  are  connected  in  an  invariable  manner  ;  and 
we  shall  then  find  it  necessary  that  the  directions  of  the  forces 
should  intersect  in  a  single  point.  For,  since  the  forces 
P  and  P'  {Pig.  46)  are  supposed  to  be  sustained  in  equilibrio 
by  the  third  force  P",  it  is  necessary  that  this  third  force 
should  be  equal  and  directly  opposed  to  the  resultant  of  the 
two  forces  P  and  P'.  But  P  and  P'  intersect  in  a  point 
D  ;  this  point  is  therefore  situated  on  their  resultant,  and 
consequently  in  the  direction  of  the  third  force  P". 

If,  on  tne  contrary,  the  force  P"  were  not  applied  at  the 
point  of  intersection  of  the  other  two,  it  would  intersect  the 
direction  of  their  resultant  R  at  some  point  E  {Pig.  47),  and 
the  right  lines  RD  and  P"E  being  then  inclined  to  each  other 

D  5 


50  STATICS. 

in  a  certain  angle  P"ER,  the  forces  R  and  P"  could  not  main- 
tain an  equilibrium  (Art.  16). 

94.  When  the  directions  of  the  three  forces  P,  P',  P"  inter- 
sect in  a  point,  this  point  may  be  considered  as  their  point  of 
application,  and  the  conditions  of  equilibrium  will  then  be 
the  same  as  if  the  forces  had  been  originally  applied  at  their 
point  of  intersection. 

These  conditions  are, 

P  cos  «  +  P'  cos  a'  +  P"  cos  «"  +  (fec.  =  0, 
P  cos  /3  +  P'  cos  /3'  +  P"  COS  |S"  +  &C.  =  0. 

To  these  must  be  added  the  equation  which  expresses  the 
condition  of  their  intersecting  in  a  point. 

95.  Let  P,  P',  and  R  {Fig.  48)  represent  three  forces  whose 
directions  intersect  at  the  point  A.  If  through  the  point  C, 
assumed  arbitrarily,  a  right  line  be  dra\\ai  to  the  point  A,  and 
perpendiculars  CI,  CI',  CI"  be  demitted  on  the  lines  of  direc- 
tion of  the  forces,  the  right-angled  triangles  CAI,  CAF,  CAI" 
will  have  the  same  hypotheneuse  CA  :  this  condition  of  a 
common  hypotheneuse  will  establish  that  of  the  forces  inter- 
secting at  a  single  point,  since  it  results  from  the  triangles 
having  a  common  vertex.  Through  the  point  A  draw  the 
right  line  AB,  perpendicular  to  CA,  and  from  the  extremities 
of  the  lines  AP,  AP',  and  AR,  which  represent  the  intensities 
of  the  forces,  demit  perpendiculars  PD,  P'D',  RD"  on  the  line 
AB  :  the  right-angled  triangles  ACI  and  APD  will  be  similar, 
having  the  alternate  angles  CAI  and  APD  equal  to  each 
other,  and  the  following  proportion  will  therefore  obtain  : 

AC  :  CI  :  :  AP  :  AD, 
and  by  calling  AC=c,  CI  =7),  this  proportion  becomes 

c : ^ : : P :  AD  ; 
whence  we  obtain 

c 

denoting  by  p'  and  r  the  perpendiculars  CI'  and  CI",  we  find, 
in  like  manner, 

c  c 

But  if  R  be  the  resultant  of  P  and  P',  the  component  of  R 


FORCES  APPLIED  TO  DIFFERENT  POINTS.       51 

in  the  direction  of  AB  will  be  equal  to  the  sum  of  the  com- 
ponents of  P  and  P'j  directed  along  the  same  line  ;  we  con- 
sequently have 

AD" = AD + AD'; 
and  by  substituting  in  this  equation  the  values  found  above, 
it  becomes 

Rr_Pp    Py 

or,  by  suppressing  the  divisor  common  to  the  terms,  it 
reduces  to 

Rr=Pp+Py (47). 

96.  If  the  point  C  were  situated  within  the  angle  formed 
by  the  directions  of  the  forces,  or  in  the  opposite  angle,  the 
product  of  the  resultant  by  the  perpendicular  r  would  then  be 
equal  to  the  difference  of  the  products  of  the  two  components 
multiplied  by  their  respective  perpendiculars  ;  we  should 
thus  have 

Rr=Pp— py (48). 

97.  The  moment  of  a  force  with  reference  to  a  plane  has 
been  defined  (Art.  81)  to  be  the  product  of  the  intensity  of 
this  force  by  the  perpendicular  on  the  plane  from  the  point 
of  application.  By  analogy,  we  call  the  moment  of  a  fmxe 
with  reference  to  a  j)oint,  the  froduct  of  the  force  hy  the  per- 
pendicular  de7nitted  on  the  direction  of  the  force  from  the 
assumed  point.  The  equations  (47)  and  (48)  will  therefore 
express  that  the  moment  of  the  resultant  of  two  forces  is 
equal  to  the  sum  or  difference  of  the  moments  of  its  compo- 
nents, according  to  the  position  of  the  point  C.  This  point  is 
called  the  centre  of  7noments  ;  and  if  it  be  situated  within  the 
angle  PAP',  or  LAL'  {Mg.  49),  the  difference  of  the  moments 
must  be  taken,  but  if  it  fall  without  these  angles,  the  moment 
of  the  resultant  will  be  equal  to  the  sum  of  the  moments. 

98.  These  two  cases  may  be  comprised  in  a  single  enun- 
ciation, by  attaching  to  the  word  sum  its  algebraic  significa- 
tion,! and  the  moment  of  the  resultant  will  then  be  equal  to 
the  sum  of  the  moments  of  the  two  components,  in  which 
expression  the  terms  may  be  affected  either  with  the  positive 
or  negative  signs.  ^ 

D2 


52 


STATICS. 


99.  The  condition  of  the  forces  intersecting  in  a  point 
gives  rise  to  the  preceding  theorem  of  the  moments  :  from 
this  theorem  the  third  condition  of  equilibrium  may  be  de- 
duced. 

For,  if  two  forces  P  and  P'  {Fig.  50)  are  sustained  in  equi- 
hbrio  by  a  third  force  P",  this  force  must  be  equal  in  intensity 
to  the  resultant  of  the  other  two,  and  must  act  in  a  direction 
exactly  opposite.  If,  therefore,  a  perpendicular  ja"  be  demitted 
on  the  line  of  direction  of  the  force  P",  which  is  also  that 
of  the  resultant  R,  the  principle  of  the  moments  will  furnish 
the  equation 

%"=:P/>  +  Py; 

and  replacing  R  by  — P",  since  the  forces  are  equal,  and  act 
in  contrary  directions,  the  equation  becomes 

Pp+py+py=o. 

Thus  the  conditions  of  equilibrium  of  three  forces  situated 
in  the  same  plane,  and  applied  to  different  points,  will  be  ex- 
pressed by  the  three  following  equations  : — 

P  cos  ^+F  cos«'  +  P"  cos  cc"=Q (49), 

P  cos/3  +  Fcos0'+P"  cos/3"=0 (50), 

P/>  +  Py +  P>"=0 (51). 

100.  If  the  number  of  forces  be  greater  than  three,  we 
may  regard  P  as  being  the  resultant  of  the  two  forces  P"'  and 
P"'  :  we  shall  then  have 

P  cos  «=P"'  cos  *"'+P'^  cos  «'", 

P  cos  /3  =  P"'  cos  /3"'  +  P^  cos  /3", 

Pp=P"y"+P'y'; 

and  by  substituting  these  values  in  equations  (49),  (50),  (51), 

they  become 

P'  cos  «'  +  P"  cos  «"  +  P"'  cos  «'"  +  P"'  cos  «"=0, 
F  cos  (3'+P"  cos  ^"  +  F"  cos  |3"'+P"  cos  ^"=0. 

V'p' + p>" + vy + Yy =0. 

101.  The  same  principle  may  be  extended  to  any  number 
of  forces,  and  we  shall  therefore  obtain  for  the  general  equa- 
tions of  equilibrium  of  forces  acting  in  the  same  plane,  and 
applied  to  different  points, 

P  cos  «  +  F  cos  a'  +  P"  cos«"  +  &c.=9 (52), 


FORCES    APPLIED   TO    DIFFERENT    POINTS.  53 

P  C0S/3  +  F  C0S/3'+P"  COS  /3"  +  <fec.=:0 (53), 

Pp  +  Py  +  Fy'  +  &c.=0 (54). 

102.  A  more  convenient  notation  is  sometimes  employed 
to  express  the  existence  of  these  conditions,  the  equations 
being  written  in  the  following  form  : — 

r(P  cos  x)=0,     2(P  cos  /3)  =0,     s(P;?)=0. 
The  character  s  is  here  employed  to  denote  the  sum  of  any 
number  of  quantities  of  the  same  form  as  those  included 
within  the  parentheses. 

103.  The  process  which  has  led  to  the  equation  (47)  fur- 
nishes an  easy  method  of  recognising  the  proper  signs  of  the 
moments.  For,  if  the  point  C,  the  centre  of  moments  {Pig. 
51),  be  chosen  without  the  angle  formed  by  the  directions  of 
the  extreme  forces,  and  the  forces  be  supposed  to  act  by  push- 
ing, being  at  the  same  time  firmly  connected  with  the  per- 
pendiculars p,  ^'^, />";  <^c.,  these  forces  will  all  tend  to  turn  the 
perpendiculars  in  the  same  direction  about  the  point  C  ;  but 
if,  on  the  contrary,  the  centre  C  be  situated  within  the  angle 
formed  by  the  directions  of  the  extreme  forces  {Pig.  52),  or 
within  the  opposite  angle,  the  forces  P,  P',  P",  &c.,  situated 
on  the  same  side  of  the  point  C,  will  tend  to  turn  the  perpen- 
diculars in  one  direction,  while  the  forces  P'",  P'',  &c.,  on  the 
opposite  side,  will  tend  to  turn  the  perpendiculars  in  a  con- 

trary  direction.     But  the  expressions  —^,  — i-,  — ^,  &c.,  rep- 

c      c         c 

resented  by  the  lines  AD,  AD',  AD",  <fcc.,  being  affected  with 

signs  contrary  to  those  of  AD'",  AD'",  (fee,  it  follows  that  all 

the  forces  whose  moments  are  positive  will  tend  to  turn  the 

system  in  one  direction,  while  those  whose  moments  are 

negative  will  tend  to  turn  it  in  a  contrary  direction.* 

*  This  demonstration  is  perfectly  conclusive  when  the  directions  of  the  several 
forces  intersect  in  a  point  ;  but  the  property  of  the  moments  is  equally  true 
when  the  forces  are  not  directed  to  a  single  point.  For,  by  prolonging  the 
directions  of  any  two  of  the  forces  P  and  P'  until  they  intersect,  and  joining 
their  point  of  intersection  with  the  centre  of  moments,  it  may  be  proved  by  the 
reasoning  employed  in  Art.  103,  that  the  moment  of  their  resultant  is  equal  to 
the  algebraic  sum  of  the  moments  of  the  two  forces  P  and  P',  the  signs  of  these 
moments  being  determined  by  the  directions  in  which  the  forces  P  and  P'  tend 
to  turn  the  system  about  the  centre  of  moments.     We  shall  thus  have 


54 


STATICS. 


104.  If  the  system  of  forces  be  not  in  equilibrio,  the 
moment  of  tlie  resultant  will  be  equal  to  the  excess  of  the 
sum  of  the  moments  of  those  forces  which  tend  to  produce 
rotation  in  one  direction,  over  the  sum  of  the  moments  of 
those  which  tend  to  turn  the  system  in  a  contrary  direction. 

105.  It  appears  from  the  preceding  remarks,  that  the 
equation  s(P/>)=0,  expresses  the  condition  that  the  sums  of 
the  moments  of  the  forces  wliich  tend  to  produce  rotation  in 
the  two  directions  are  equal  to  each  other. 

106.  If,  in  the  system  supposed  in  equilibrio,  we  suppress 
one  of  the  components,  P  for  example,  the  remaining  forces 
will  have  a  resultant  R  ;  and  since  this  resultant  should  be 
equal  in  intensity,  but  directly  opposed  to  the  force  P,  the 
equations  (52),  (53),  and  (54)  will  be  replaced  by  the  following: 

R  cos  a=F'  cos  a'  +  P"  cos  a"  +  F"  cos  «"'4-&c., 
R  cos  b^V  cos  /3'  +  P"  cos  /3"  +  P'"  cos  /3"'  +  (fcc, 
Rr = Vy + V"p" + P'>'" + &c.  ; 
or, 

R  cos  «  =  2(P  cos  et)  =X, 

R  cos  b  =!:.(?  cos  /3)=Y, 
Rr=s(P/>). 


The  double  sign  is  not  prefixed  to  the  moment  P;?,  since  we  are  at  liberty  to 
assume  arbitrarily  the  sign  of  one  of  the  moments.  The  moment  Rr,  deduced 
from  this  equation,  may  have  either  a  positive  or  negative  value  ;  if  positive,  R 
and  P  will  tend  to  turn  the  system  in  the  same  direction  ;  if  negative,  in  con- 
trary directions. 

The  forces  P  and  P',  being  then  replaced  by  their  resultant  R,  this  resultant 
can  be  combined  with  a  third  force  P",  and  we  shall  obtain,  in  a  similar  maruier, 
R'r'=Rr±PY'; 

in  which  equation  Rr,  whatever  may  be  its  essential  sign,  may  be  replaced  by 
P^^Py.  The  sign  of  the  moment  P"p"  will  be  similar  to  that  of  Rr,  if  P" 
and  R  tend  to  produce  rotation  in  the  same  direction,  and  dissimilar  in  the  con- 
trary case.  But  the  moments  Pj)  and  Rr  will  have  like  or  unlike  signs,  according 
as  the  forces  P  and  R  tend  to  turn  the  system  in  the  same  or  in  contrary  direc- 
tions. Hence  the  signs  of  the  moments  Vp  and  P"p"  in  the  equation  R'r'= 
T'pizP'p':izP"p'\  will  be  like  or  unlike  according  to  the  directions  in  which  the 
forces  P  and  P"  tend  to  produce  rotation. 

The  same  reasoning  may  be  extended  to  a  greater  number  efforces. 


FORCES  APPLIED  TO  DIFFERENT  POINTS.       55 

107.  By  means  of  these  equations,  the  position  and  mag- 
nitude of  the  resultant  may  be  determined. 

For,  the  two  first  equations  give 

R2(cos2a  +  cos26)=X2  +Y2  ; 

and  since  the  sum  of  the  squares  of  the  two  cosines  is  equal 
to  unity,  we  have 

R2=X2+Y2. 

The  inchnations  of  the  resultant  to  the  co-ordinate  axes  may 
also  be  determined  from  the  same  equations  ;  for  we  have 

X  Y 

cos  a=— ,     cos  6  =-5-. 
K  Jti 

108.  To  establish  its  position  in  the  system,  we  first  deter- 
mine the  position  of  a  right  line  AB,  passing  through  the 
origin,  and  parallel  to  the  resultant.  If  cos  b  be  affected 
with  the  positive  sign,  the  line  AB  must  form  with  the  axis 
of  y  an  angle  less  than  90°  :  it  will  therefore  assume  one  of 
the  positions  indicated  in  {Fig.  53).  But  if,  on  the  contrary, 
this  quantity  should  have  the  negative  sign,  the  right  line  AB 
would  then  be  situated  in  one  of  the  positions  represented  by 
{Pig.  54).  Thus,  whatever  be  the  sign  of  cos  6,  the  line  AB 
may  assume  two  positions,  one  in  which  the  angle  formed 
with  Kx  will  be  obtuse,  and  another  in  which  this  angle  will 
be  acute.  The  sign  of  the  cos  a  will  determine  which  of 
these  positions  the  line  AB  must  assume. 

Having  thus  established  the  position  of  the  right  line  AB,  let 
a  perpendicular  /■  be  drawn  to  it  through  the  origin  A,  equal  to 

s(P«) 
ji   •     This  perpendicular  will  be  represented  {Pig.  55)  by 

AO  or  by  AO',  according  to  the  sign  of  the  quantity  r  ;  and 
the  line  OR  or  O'R',  parallel  to  AB,  will  represent  the  true 
position  of  the  resultant. 

109.  To.  obtain  the  equation  of  this  resultant,  it  may  be 
observed  that  its  line  of  direction  will,  in  general,  intersect 
the  axis  of  y  at  a  certain  point  B  {Pig.  56),  and  that  the  form 
of  its  equation  will  therefore  be 

y=x  tang  D-f  AB (55)  ; 


56  STATICS. 

and  since  the  angle  which  the  resultant  makes  with  the  axis 
of  a;  is  denoted  by  a,  we  have  T>=a,  and  consequently 

-r,    sin  a     cos  b     R  cos  b    Y 

tanff  D= = =c; =-— . 

cos  a     cos  a    K  cos  a    X 

The  value  of  AB  may  be  obtained  from  the  equation 

OA=ABxcosOAB. 
But  the  angle  OAB  is  equal  to  the  angle  D,  since  they  are 
both  complements  of  OAD.  The  angle  OAB  can  therefore 
be  replaced  in  the  preceding  equation  by  D  or  a  ;  and  since 
the  line  AO  is  the  perpendicular  from  the  origin  on  the  direc- 
tion of  the  resultant,  it  will  represent  the  quantity  denoted 
by  r  ;  we  shall  thus  obtain 

?'=AB  cos  a  ; 
and  consequently, 

AB=-!1-. 

cos  a 

Substituting  the  values  of  AB  and  tang  D  in  the  general 
equation  (55),  it  becomes 

Y         r    _Y  Rr__Y      Rr. 

'^"X'^'^co^ ~ X'^'^Rc^i^^X''^'^ X  ' 

whence,  by  transposition  and  reduction,  we  find 

yX-a:Y=Rr] 
or,  replacing  Rr  by  its  equal  ^(Pj)),  the  equation  of  the 
resultant  finally  becomes 

yX-xY=i:(Pp). 

110.  When  an  equilibrium  subsists,  X  and  Y  are  equal  to 
zero,  and  the  equation  reduces  to  s(P/?)=0,  corresponding 
with  the  result  previously  obtained. 

111.  The  data  requisite  for  the  determination  of  the 
resultant  being,  1°.  The  intensities  of  the  several  forces;  2°. 
Tile  angles  on  which  their  dii'eciions  depend  ;  and  3".  The 
co-ordinates  of  their  points  of  application,  it  will  prove  con- 
venient to  transform  the  equation  (54)  into  another,  in  which 
the  quantities  />,  p',  p",  ôcc.  shall  be  replaced  by  the  co-ordi- 
nates of  the  points  of  application.  To  effect  this  transforma- 
tion, let  the  origin  of  co-ordinates  be  assumed  at  A  {Fig.  57), 


FORCES    APPLIED   TO    DIFFERENT    POINTS.  57 

and  let  x  and  y  denote  the  co-ordinates  of  the  point  M  to 
which  a  force  P  is  applied  :  the  intensity  of  this  force  being 
represented  by  MP,  its  components  parallel  to  the  axes  of  x 
and  y  will  be  respectively 

MN=P  cos  «, 

MQ,=P  cos  /3. 

From  the  point  A  demit  the  perpendiculars  AO,  AF,  and  AE 

on  the  prolongations  of  the  force  MP  and  its  two  components  j 

we  shall  then  have 

OAxMP=the  moment  of  the  force  P, 
AF  X  MN=the  moment  of  the  component  P  cos  «, 
AE  X  MQ,=the  moment  of  the  component  P  cos  j3. 
But  if  we  regard  the  forces  as  pushing  the  point  M,  the 
resultant  MP  and  the  component  P  cos  «  will  tend  to  produce 
rotation  in  the  same  direction  about  the  point  A.     Their 
moments  may  therefore  be  alfected  with  the  positive  sign  ; 
while  the  component  P  cos  /3,  tending  to  turn  the  system  in  a 
contrary  direction,  must  be  affected  with  the  negative  sign. 
We  shall  thus  obtain  the  equation 

Pp=yP  cos  » — xY  cos  /3. 
For  a  similar  reason, 

P'jo'=3/'F  cos  «'-.r'F  cos  /3', 

P"p"=y"P"  cos  a:'—x"Y"  COS  /3"j 

<fcc.  &c.  <fcc.  ; 

and  by  substituting  these  values  in  the  equation  of  the  mo- 
ments (54),  it  becomes 

P(y  cos  tt—x  cos  |3)  +  P'(y'  cos  «' — x'  cos  /s')  +  &c.  =0  . . . .  (56)  t 
we  shall  therefore  have  for  the  equation  of  the  resultant,  when 
the  system  is  not  in  equilibrio  (Art.  109), 

2/X — a:Y=s[P(2/  cos  a— a:  cos/3)]. 
112.  Jn  determining  the  signs  of  the  moments  in  equation 
(54),  we  had  recourse  to  the  rule  explained  in  Art.  103,  which 
is  somewhat  foreign  to  analytical  considerations  ;  but  when, 
by  a  transformation,  this  equation  takes  the  form  indicated 
above  (56),  the  signs  of  the  moments  will  be  immediately 
determined  by  an  application  of  the  rule  in  Arts.  37  and  38, 
regard  being  had  to  the  signs  of  the  co-ordinates.     Thus,  let 


68  STATICS. 

P  be  a  force  whose  position  with  respect  to  the  co-ordinate 
axes  is  that  represented  in  {Fig.  58).  The  value  of  its  mo- 
ment, being  in  general  V{i/  cos  « — x  cos/î),  will  become  appli- 
cable to  the  particular  case,  by  making  x  negative,  y  positive, 
cos  X  negative,  cos  p  negative  :  thus,  when  the  signs  are  con- 
sidered, the  moment  becomes 

P( — y  cos  x—x  cos  |8). 

113.  It  should  be  remarked,  however,  that  we  here  adopt 
tacitly  an  hypothesis  relative  to  the  signs,  which  consists  in 
regarding  a  moment  as  positive,  when  the  direction  of  the 
force  CD  {Mg.  57)  intersects  the  axis  of  y  positive,  and  then 
cuts  the  axis  of  x  negative. 

114.  The  equations  of  equilibrium  (49),  (50),  and  (51) 
imply  the  condition  that  the  system  may  be  reduced  to  two 
forces  equal  in  intensity  and  directly  opposite.  For,  if  we 
denote  by  P  cos  «,  P'  cos  «',  &c.  the  components  acting  in  one 
direction  parallel  to  the  axis  of  x,  and  by  P"  cos  «",  P'"  cos  «"', 
&c.  the  components  which  act  in  a  contrary  direction,  the 
equation  (49)  may  be  put  under  the  form 

P  cos  x  +  V  cos  «'-f  &C.=P"  cos  x"  +  F"'  cos  a'"  +écC. 

But  the  forces  P  cos  «,  P'  cos  «',  &c.,  being  parallel,  may  be 
compounded  into  a  single  force  X',  equal  to  their  sum  and 
parallel  to  them  ;  and  the  forces  P"  cos  «",  P'"  cos  a",  <fec. 
may  in  like  manner  be  replaced  by  a  single  force  X"  :  the 
entire  system  will  thus  be  reduced  to  the  two  forces  X'  and 
X",  parallel  and  equal,  but  having  contrary  directions. 

By  a  similar  composition,  the  forces  parallel  to  the  axis  of  y 
may  be  reduced  to  two  resultants  Y'  and  Y",  equal  to  each 
other,  and  having  opposite  directions. 

The  forces  X'  and  Y'  being  then  applied  at  the  point  M, 
where  their  directions  intersect  {Pig.  59),  and  the  forces 
X"  and  Y"  at  their  point  of  intersection  N,  we  can  construct 
the  rectangles  MA  and  NB,  whose  sides  MC,  MD,  NE,  and 
NF  shall  represent  the  forces  X',  Y',  X",  Y"  :  and  since  the 
homologous  sides  of  these  rectangles  are  equal,  their  diago- 
nals MA  and  NB  will  also  be  equal  and  parallel. 

The  equations  X=0,  Y =0,  therefore,  express  that  forces 
fiituated  in  a  plane  may  be  reduced  to  two  INI  A  and  NB,  equal, 


FORCES    APPLIED   TO    DIFFERENT    POINTS.  &» 

parallel,  and  acting  in  contrary  directions  ;  but  they  do  not 
express  the  condition  that  the  two  forces  are  directly  opposed. 
That  this  may  occur,  the  equation  s(P^)=:0  is  likewise  neces- 
sary :  for,  calling  R'  and  R"  the  two  equal  forces  AM  and  BN, 
and  /•',  r"  the  perpendiculars  OP  and  OGl  demitted  from  the 
point  O,  since  R'  and  R"  act  in  contrary  directions,  their  mo- 
ments must  be  taken  with  différent  signs,  and  the  equation 
s(Pp)  =0  will  be  replaced  by  the  following  : 

RV— R"r"=0. 
But  the  intensities  of  R'  and  R"  being  equal  by  hypothesis,  the 
common  factor  will  disappear  from  the  equation,  and  it  will 
then  become 

r'— r"=0; 
thus,  the  difference  of  the  right  lines  OP  and  OQ,  will  become 
equal  to  zero,  and  the  points  P  and  Q,  will  therefore  coincide  : 
hence,  the  forces  MA  and  NB  will  be  directed  along  the  same 
right  line. 

It  also  appears  that  when  the  condition  2(Pp)  =0  is  not 
fulfilled,  and  we  have  simply  X=0,  Y=0,  the  system  may 
be  reduced  to  two  parallel  forces  similarly  situated  to  those 
considered  in  Art.  74. 

115.  If,  on  the  contrary,  the  condition  s(P/>)=0  were  alone 
satisfied,  an  equilibrium  could  not  subsist  ;  for  the  quantities 
X  and  Y  having  certain  values,  a  resultant  might  be  found 
whose  intensity  would  be  determined  by  means  of  the 
equation 

In  this  case,  the  equation  s(P/»)=0,  or  its  equivalent  Rr=0, 
can  only  be  satisfied  by  making  the  factor  r  equal  to  zero  ; 
hence,  the  centre  of  moments  must  necessarily  be  found  on 
the  line  of  direction  of  the  resultant  R. 

116.  If  there  be  a  fixed  point  on  the  line  of  direction  of  the 
resultant,  the  equilibrium  will  be  still  maintained,  and  the 
centre  of  moments  being  placed  at  this  point,  the  condition 
2(P/))  =0  will  be  satisfied  ;  if,  for  example,  the  forces  P,  P',  P'^^ 
&,c.  be  supposed  applied  to  the  different  points  of  a  solid  body, 
and  if  the  point  C  through  which  the  resultant  passes  be  im- 
moveable, the  effect  of  this  resultant  will  be  entirely  destroyed 


60  STATICS. 

by  the  reaction  of  the  fixed  point,  and  the  condition  2(Py>)  =0 
will  be  alone  sufficient  to  ensure  the  equilibrium.  It  will 
appear  hereafter  that  the  intensity  of  this  resultant  is  a 
measure  of  the  pressure  sustained  by  the  fixed  point. 

117.  If  the  system  can  be  reduced  to  two  parallel  forces, 
equal  in  intensity,  but  not  directly  opposed,  the  addition  of  an 
arbitrary  force  S  will  render  it  susceptible  of  a  single  result- 
ant. For  the  new  force  S  must  necessarily  be  either  par- 
allel or  inclined  to  the  direction  of  the  forces  ;  in  the  first  case 
{Fig.  60),  it  may  be  decomposed  into  two  parallel  components 
P'  and  Q.'  applied  at  the  points  A  and  B  (Art.  73),  and  the  sys- 
tem of  three  forces  P,  Q,,  and  S  will  be  replaced  by  the  two 
unequal  forces  P  +  P'  applied  at  A,  and  Q, — Q.'  applied  at  B  ; 
these  two  forces  will  obviously  have  a  single  resultant. 

If  the  new  force  S  is  not  parallel  to  the  other  two,  its 
direction  may  be  prolonged  {Fig.  61)  until  it  intersects  the 
direction  of  one  of  them  at  A'.  This  point  being  then  taken 
as  the  point  of  application  of  the  forces  P  and  S,  they  may  be 
compounded  by  constructing  a  parallelogram  on  their  lines 
of  direction,  and  the  direction  of  their  resultant  will  intersect 
that  of  the  force  Q,  with  which  force  this  resultant  may  be 
combined. 

Of  Forces  acting  in  any  tnanner  in  Space. 

118.  Let  P',  P",  P'",  (fcc.  represent  different  forces  situated 
in  space  ; 

x',  2/,  z',  the  co-ordinates  of  the  point  of  application  of  P', 

x'^,  y'\  z",  those  of  P", 

x"',  y'",  z"\  those  of  P'", 
&c.     &.C.     &c.  ; 
ce'j  /3',  y',  the  angles  formed  by  P'  with  the  axes  of  co-ordinates, 
tt\  /3", y'',  those  formed  by  P"  with  the  axes, 
«'",  0'",  y'",  those  formed  by  P'"  with  the  axes, 

&c.  ■     (fcc.  (fee. 

Let  us  investigate  the  conditions  of  equilibrium  iw  this  sj'-s- 
tem,  and  endeavour  to  discover  if  these  conditions  cannot  be 


Ïi-ORCES   ACTING    IN   STfACE.  61 

rendered  dependent  on  those  which  have  been  obtained  in 
the  preceding  cases.  We  first  attempt  to  decompose  all  the 
forces  of  the  system  into  two  groups,  one  of  which  shall  con- 
sist of  parallel  components,  and  the  second  of  forces  situated  in 
the  same  plane.  Since  the  axes  of  co-ordinates  may  be  as- 
sumed arbitrarily,  we  will  endeavour  to  decompose  the  forces 
in  such  manner  that  a  certain  number  of  them  may  be  in 
the  plane  of  a;,  y,  and  the  remainder  be  parallel  to  the  axis 
of  z. 

119.  If  in  the  given  system  there  be  no  force  parallel  to 
the  plane  of  a;,  y,  the  proposed  decomposition  may  be  readily 
effected  ;  for,  let  one  of  the  forces  be  represented  by  P',  its 
point  of  application  being  at  W  {Fig.  62);  prolong  the  hne 
of  direction  of  this  force  until  it  intersects  at  C  the  plane  of 
.r,  y,  and  transferring  the  point  of  application  to  C,  decom- 
pose the  force  P'  into  two  others,  one  C'L  parallel  to  the  axis 
of  2:,  the  other  C'N  in  the  plane  of  .r,  y. 

120.  But  if  the  force  P'  is  parallel  to  the  plane  of  a:,  y,  a 
similar  decomposition  cannot  be  effected,  and  some  other 
mode  of  decomposing  the  forces  must  therefore  be  adopted. 

For  this  purpose,  let  there  be  drawn  through  the  point  M' 
{Fig.  63)  a  line  parallel  to  the  axis  of  2;,  and  to  the  point  M' 
let  there  be  applied  along  this  line,  and  in  contrary  directions, 
the  two  forces  M'O  and  M'O',  having  intensities  equal  to  g' 
and  —g'  respectively.  The  introduction  of  these  forces 
cannot  disturb  the  condition  of  the  system,  since  the  two 
mutually  destroy  each  other  ;  and  we  shall  then  have  applied 
at  the  point  M'  the  three  forces  P',  g\  and  — g'. 

The  force  P'  may  then  be  compounded  with  — g\  and  by 
calling  their  resultant  R',  we  can  replace  in  the  system  the 
force  P',  by  the  two  forces  R'  and  g\  each  of  which  must  ob- 
viously intersect  the  plane  of  a:,  y. 

121.  Let  the  force  R'  be  now  applied  at  C,  the  point  in 
which  its  line  of  direction  intersects  the  plane  of  a:,  y,  and  let 
it  be  decomposed  into  two  components,  one  situated  in  the 
plane  of  .r,  y,  and  the  other  parallel  to  the  axis  of  z.  The 
force  P'  will  thus  be  replaced  by  a  force  applied  at  C,  and 
lying  in  the  plane  of  a:,  y,  and  by  two  others  parallel  to  the 
axis  of  2;,  one  applied  at  C,  and  the  other  at  M'. 

122.  The  co-ordinates  of  the  points  of  application  being 

6 


62 


Î3TATICS. 


necessary  to  express  the  conditions  of  equilibrium,  those  of 
the  point  C  must  be  determined. 

The  equations  of  the  resultant  R'  which  passes  through  the 
point  .t',  y',  2;',  have  been  found  (Art.  57)  to  be  of  the  form 


Z 

z — z'=^{x — x') 


(57); 


in  which  X,  Y,  and  Z  represent  the  projections  of  R'  on  the 
co-ordinate  axes.  These  projections  being  equal  to  the  com- 
ponents of  R'  parallel  to  the  axes,  the  quantities  X,  Y,  and  Z 
may  be  replaced  by  the  values  of  the  three  components.  But 
R'  being  the  resultant  of  P'  and— ^',  we  may  substitute  for 
F  its  three  components  F  cos  «',  P'  cos  /S',  P'  cos  y'  ;  and  R'^ 
will  then  be  the  resultant  of  the  four  forces 

P'  cos  a,      P'  cos  /3',      F  cos  y',       —g^. 

These  forces  acting  parallel  to  the  axes  of  co-ordinates,  we 
shall  have 

X=F  cos  «',    Y=F  cos  /3',    Z=F  cos  y'-g'; 
and  by  substituting  these  values  in  equations  (57),  we  obtain 
for  the  equations  of  the  resultant  R', 


,     P'  cos  y'  —ff,         ,v     1 
F  cos  «'    ^  I 

Fcosy'-^,         ,-     ^ 


(58). 


F  cos  /3' 

123.  To  obtain  the  co-ordinates  of  the  point  C  (Pig.  63), 
at  which  the  right  line  R'  intersects  the  plane  of  x,  y,  we 
make  z=0  in  the  equations  (58)  ;  and  denoting  by  a,  and  6, 
the  other  two  co-ordinates  of  the  point  C,  we  shall  have 
^,_Fcosy'— ^' 


—z' 


Z=:- 


P'  cos  «' 
P'cos  y' — £r' 


F  cos  /3' 


(a—x'), 

(i-yO; 


from  which  we  deduce 


a  =x 


h,=t/' 


Z'V  cos  a' 

P'  cos  y' — g' 

z'V  cos  iS' 
P'  cos  y'—g" 


(59): 


FORCES  ACTING  IN  SPACE.  63 

these  are  the  values  of  the  co-ordinates  of  the  point  C,  at 
which  the  resultant  R'  intersects  the  plane  of  x,  y. 

124.  The  force  R',  being  represented  in  intensity  by  the 
line  M'R'  {Fig.  64),  may  be  supposed  applied  at  C,  in  its  line 
of  direction.  Then  making  CD' = M'R',  and  decomposing 
CD'  into  three  rectangular  forces,  applied  at  C  and  parallel 
to  the  co-ordinate  axes,  these  components  will  be  equal  to 
those  of  the  force  M'R'  ;  and  the  point  C  may  therefore  be 
considered  as  solicited  by  the  three  forces  F  cos  «',  P'  cos  ^\ 
and  P'  cos  y'—g\  the  two  former  being  situated  in  the  plane 
of  a-,  y,  and  the  latter  parallel  to  the  axis  of  z.  Thus,  instead 
of  the  force  P'  applied  at  M',  we  shall  have 

the  force  g  applied  at  M',  parallel  to  the  axis  of  z, 
the  force  P'  cos  y'—g  applied  at  C,  parallel  to  the  axis  of  Zy 
the  force  P'  cos  »  applied  at  C,  and  acting  in  the  plane  of  ar,  y, 
the  force  P'  cos  ^  applied  at  C,  and  acting  in  the  plane  of  x,  y. 

125.  By  adopting  a  similar  method  of  decomposition  for 
the  forces  P",  P  ",  &c.,  employing  the  auxiliary  forces  g'\  g"\ 
&c.,  applied  at  the  points  M",  M",  (fee,  the  system  will  be 
reduced  to  two  groups  of  forces,  of  which  one  will  have  its 
components  parallel  to  the  axis  of  z,  and  the  other  will  be 
situated  in  the  plane  of  x,  y. 

The  forces  parallel  to  the  axis  of  z  will  be 

g\  g'\  g"\  &c., 
applied  at  the  points  M',  M",  M'",  «fcc.  ;  and 

P'  cos  y'—g,     P"  cos  y"—g'\     F"  cos  y"'—g"',  &c., 
applied  at  the  points  C,  C",  C",  &c. 

And  the  forces  lying  in  the  plane  of  x,  y,  will  be 
P'  cos  «',    F'  cos  «",    F"  cos  «'",  (fee, 
applied  at  the  points  C,  C  ",  C  ",  &c.  ^  and 

P'  cos  (3',      P"  COS  fl'',      P'"  COS  /3"',  &C. 

applied  at  the  same  points  C,  C",  C  ",  (fcc. 

126.  It  will  now  be  demonstrated  that  when  an  equilibrium 
subsists  in  the  system,  it  will  be  necessary,  1°.  that  the  forces 
parallel  to  the  axis  of  z  should  be  in  equilibrio  ;  2".  that  the 
forces  acting  in  the  plane  of  a:,  y,  should  also  destroy  each 
other. 


64  STATICS. 

For  since  the  equilibrium  is  supposed  to  subsist,  the  state 
of  the  system  will  not  be  changed  by  supposing  a  line  C'C" 
assumed  arbitrarily  in  the  plane  of  x,  y  {Fig:  65)  to  become 
immoveable.  The  forces  situated  in  this  plane  will  then  be 
destroyed  by  the  resistance  of  the  fixed  line.  For,  every 
force  in  the  plane  of  .r,  y  must  intersect  the  fixed  line,  or  be 
parallel  to  it.  In  the  first  case,  let  the  force  be  represented 
by  AB,  and  prolong  its  line  of  direction  until  it  intersects  the 
fixed  line  at  a  point  O  :  this  point  being  supposed  immove- 
able, the  effect  of  the  force  AB,  which  is  transmitted  to  the 
point,  must  be  destroyed.  Again,  if  the  force  be  parallel  to 
the  line  C'C  ",  its  point  of  application  E  cannot  be  moved 
without  communicating  a  motion  to  the  line  C'C"  which  by 
hypothesis  is  immoveable.  The  effect  of  this  force  must 
therefore  be  destroyed  by  the  fixed  line.  Thus,  the  forces 
lying  in  the  plane  of  x,  y  being  destroyed,  the  system  will  be 
reduced  to  the  group  parallel  to  the  axis  of  z.  These  latter 
forces  would  obviously  tend  to  turn  the  system  about  the 
fixed  line  C'C,  unless  the  forces  should  be  in  equilibrio,  or 
their  resultant  should  pass  through  the  fixed  line.  But  the 
position  of  this  line  having  been  assumed  arbitrarily,  it  cannot 
happen  that  the  resultant  of  the  forces  parallel  to  the  axis  of 
z  will  always  pass  through  this  line.  These  parallel  forces 
must  therefore  be  in  equilibrio. 

The  group  parallel  to  the  axis  of  z  being  in  equilibrio,  the 
forces  lying  in  the  plane  of  x,  y  must  mutually  destroy  each 
other,  since  the  equilibrium  of  the  entire  system  could  not 
oth-erwise  be  preserved. 

127.  The  problem  is  thus  reduced  to  finding  the  conditions 
of  equilibrium,  1°.  of  a  system  of  fr  .-_es  parallel  to  the  axis  of 
z\  2°.  of  the  forces  acting  in  the  plane  of  x,  y. 


Conditions  of  Equilibrmm  of  the  Forces  parallel  to  the 
Axis  of  z. 

128.  These  conditions  being  the  same  as  those  enunciated 
in  Art.  87,  the  following  quantities  must  be  equal  to  zero, — 


FORCES  ACTING  IN  SPACE.  65 

1°.  The  sum  of  the  forces  parallel  to  the  axis  of  z  ; 

2°.  The  sum  of  the  moments  taken  with  reference  to  the 

plane  of  y,  ^  ; 
3°.  The  sum  of  the  moments  taken  with  reference  to  the 

plane  oi  x\  z. 
The  first  of  these  conditions  gives 

F  cos  y—g'+g'J^Y'  cos  y"—g"+g" 
+ P"'  cos  y"  —g" -]rg"'  +  «Sec.  =  0  ; 
or,  by  reduction, 

F  cos  y'  +  P"  cos  y"  +  F"  COS  y"'  +  &C.=0 (60). 

The  second  condition  requires  the  consideration  of  two  dif- 
ferent sets  of  moments. 

1°.  Those  of  the  forces  g\  g'\  g"\  &c.,  applied  at  the  points 
M',  M",  M'",  &c. 

2°.  Those  of  the  forces  P'  cos  y  —g'^  P"  cos  y'—g'\ 
P"'  cosy'"— ^"',  (fcc,  applied  at  the  points  C,  C",  C",  (fee. 

The  moment  of  the  force  g'  applied  at  M'  {Pig.  66), 
taken  with  reference  to  the  plane  of  y,  z^  is  g  X  M'N'  :  but 
M'N'=B'D'=.t''  ;  the  moment  therefore  becomes  g'x. 

The  moment  of  the  force  F  cos  y — g  applied  at  C,  taken 
with  reference  to  the  same  plane,  is  evidently  (F  cos  y — g'") 
XE'C,  or  (P'  cosy— ^')a,  ;  and  the  sum  of  the  moments  of 
the  two  forces  will  therefore  be  represented  by 
g-'x'  +  (P'  cos  y'—g')a,. 

Substituting  in  this  expression  the  value  of  a,  (59)  deter- 
mined in  Art.  123,  we  obtain 

gx^(^  cosy— ^)  I  X'— ,5; — -1; 

\  P  COSy— ^/ 

performing  the  multiplications  indicated,  and  reducing,  we  get 

.-pT'cosy'— «T'cos»'. 
By  a  similar  process,  the  moments  of  the  parallel  forces 
applied  at  M",  M'",  C",  C",  (fee.  may  be  obtained,  and  being 
collected  into  one  sum,  the  equation  expressing  the  second 
condition  of  equilibrium  becomes 

Y{x  cos  y — 2;'  cos  «')  •\-Y\x"  cos  y — 2;"  cos  «") 

+P'"(a:"'  cos  y"'—z"  cos  «"')  +(kc.=0 (61). 

To  obtain  the  third  condition  of  equilibrium  of  parallel! 

E 


66  STATICS. 

forces,  we  find  tlie  moment  of  the  force  g'  applied  at  M', 
talccn  with  reference  to  the  plane  of  a;,  z,  and  that  of  the  force 
P'  cos  y — g'  applied  at  C,  taken  with  reference  to  the  same 
plane  :  the  first  of  these  will  be  equal  to  ^'xM'L'=^'  xB'G' 
=g  y^y  ;  the  second  will  be  (P'  cosy' — g')h^  ;  and  their  sum 
will  be  expressed  by 

^y+(Fcosy'-^')5,. 
Substituting  for  b,  its  value  (59)  found  in  Art.  123,  and  re- 
ducing, we  obtain 

yV  cos  y' — z'P'  cos  /3'. 
And  by  finding  the  moments  of  the  other  parallel  forces, 
taken  with  reference  to  the  plane  of  x,  z^  we  shall  have  for 
the  third  condition  of  equilibrium, 

F(2/'  cos  y'—Z  cos  /S)  +P  "(2/''  cos  y"—z"  COS  |3") 

+F"(y"  cosy'"— ^"'  cos/î"')+&c.=0 (62). 

Conditions  of  Equilibrium  of  the  Forces  situated  in  the 
Plane  of  x,  y. 

129.  These  conditions  being  such  as  arise  when  the  forces 
act  in  the  same  plane,  it  is  necessary, 

1°.  That  the  sum  of  the  components  parallel  to  the  axis 
of  X  should  be  equal  to  zero. 

2°.  That  the  sum  of  the  components  parallel  to  the  axis 
of  y  should  be  equal  to  zero. 

3°.  That  the  sum  of  the  moments  of  the  forces  taken  with 
reference  to  the  origin  should  be  equal  to  zero. 

The  first  two  conditions  are  expressed  by  the  equations, 

F  cosa'+P"  cos  «"  +  P'"  cos«'"+(fcc.=0 (63), 

P'cos  /3'  +  P"  C0S/3"  +  F"  cos/3"'  +  &c.=0 (64). 

"With  regard  to  the  third,,  it  may  be  observed,  that  the  two 
forces  F  cos  a!  and  P'  cos  /3'  are  applied  at  the  point  C  {Fig- 
67)  ;  the  moment  of  the  fir«tj  being  taken  with  reference  to 
the  origin  A,  will  be 

F  cos  «'  X  AE'=F  cos  «'  X  C'F'=F  cos  «' .  6,  ; 
in  like  manner,  the  moment  of  the  force  P'  cos  /3',  taken  with 
reference  to  the  origin  A,  will  be 

F  cos  ^'XAF'=:F  cos  p'xE'C'^rF  cos/3',  a,. 


FORCES-  ACTING  II»  SPACE.  67 

These  moments  should  be  taken  with  contrary  signs,  since 

the  two  components  P'  cos  «'  and  P'  cos  /s'  tend  to  turn  the 

system  in  contrary  directions  about  the  point  A.     Thus,  by 

regarding  that  momeni  as  positive  in  which  the  component 

P'  cos  «'  enters,  the  sum  of  the  moments  may  be  written 

F  cos  u  X  b^—V  cos  /3'  X  a,  ; 

substituting  in  this  expression  the  vahies  of  a,  and  Z>,  (59), 

we  get 

•n,          >  t  '        ^'P'  cos  /3'   \      ^,          ,/  ,      z'Y  cos  <*'  \ 
F  cos  a!  (  y  — _ —'—  )  — P  cos  /3  (  x'—^, -,  )  ; 

\  PcoSy— o-/  V  Pcosy— ^/ 

and  by  performing  the  multiphcations,  and  reducing,  we- 
obtain 

y'Y  cos  x—x'Y  cos /3'. 

The  moments  of  the  forces  apphed  at  C",  C",  &c.,  being  found 
in  a  similar  manner,  the  third  condition  of  equilibrium  of  the 
forces  which  lie  in  the  plane  of  x,  y  becomes 

F(y'  cos  «' — x  cos /3')  +  P"(y"  cos  x' — x" cos /3") 
+  P"'(2/"'  COSa:"—x"  COS|3"')  +  (fec.=0 (65). 

130.  The  six  equations  of  equilibrium  (60),  (61),  (62), 
(63),  (64),  (65),  may  be  written  under  the  following  form  : 

2(P  cos«)=0  ^ 

x(P  cos /3)  =0  V  . (66). 

2(Pcosy)=0> 

5:[F(y  cos  at, — X  cos  (S)]  =0  ^ 

s[P(x  cosy — z  cos«)]=0  > (67). 

2[P(y  cos  y — z  cos  /3)]  =0  3 

131.  If  there  be  a  fixed  point  in  the  system,  the  six  equa- 
tions will  not  be  requisite  to  express  the  conditions  of  equi- 
librium. For,  if  the  origin  be  placed  at  the  fixed  point,  the 
equilibrium  will  subsist  between  the  forces  acting  in  the  plane 
of  X,  y.  when  the  system  has  no  tendency  to  turn  about  this 
point.     This  condition  will  be  fulfilled  when  we  have 

2[P(2/  cos  « — X  cos  /3)J=.0. 
It  remains  to  discover  the  conditions  of  equilibrium  of  the 
forces  parallel  to  the  axis  of  z.     Let  .t„  y,,  and  0  be  the  co- 
ordinates of  the  point  at  which  the  resultant  of  the  parallel 
forces  intersects  the  plane  of  a-,  y  ;  the  moment  of  this  result- 

E2 


68  STATICS. 

ant  taken  with  reference  to  the  planes  of  x,  z,  and  y,  z,  will  be 
equal  to  the  sum  of  the  moments  of  the  several  forces  taken 
with  reference  to  the  same  planes  ;  whence  we  have 

'Rx=-l\P{x  cosy— 2;  COS  a)], 
Ry^  =  2.[P(y  COS  y — Z  COS  /s)]. 

If  an  equilibrium  subsists  between  the  parallel  forces,  their 
resultant  must  pass  through  the  fixed  point,  which,  by  hy- 
pothesis, coincides  with  the  origin  of  co-ordinates,  and  we 
therefore  have  :r,=0,  y=0.  The  preceding  equations  will 
thus  be  reduced  to 

Y.\P{X  cos  y — Z  cos  «)J=0, 

5;[P(y  cos  y—z  cos/3)]=0. 
We  therefore  conclude  that  when  the  system  contains  a  fixed 
point,  the  equilibrium  will  subsist,  if  the  equations  (67)  are 
alone  satisfied,  the  origin  being  taken  at  the  fixed  point. 

132.  Wlien  the  system  contains  two  fixed  points,  one  of 
the  co-ordinate  axes  may  be  drawn  through  them  ;  this  axis 
will  thus  become  fixed,  and  the  system  can  only  be  subject  to 
a  motion  around  it.  A  similar  case  will  be  examined  in  the 
succeeding  paragraph. 

133.  When  there  exists  a  fixed  axis  about  which  the  system 
may  turn,  this  axis  may  be  assumed  as  the  axis  of  z,  and  the 
forces  parallel  to  it  will  produce  no  effect.  The  remaining 
forces  are  situated  in  the  plane  of  x,  y.  But  the  condition  of 
equilibrium  of  these  forces  requires  that  their  resultant  should 
pass  through  the  point  A  {Fig.  67),  which  point  is  immove- 
able, being  on  the  axis  of  z  ;  and  the  condition  of  the  result- 
ant'spassing  through  A  is  expressed,  as  above,  by  the  equation 

2:[P(y  cos  a—x  cos  /3)]=0. 
This  equation  expresses  that   the    system  is  in  equilibrio, 
when  the  axis  of  z  is  supposed  fixed. 

134.  If  we  suppose,  successively,  the  axes  of  y  and  x  to 
become  fixed,  it  may  in  like  manner  be  demonstrated  that 
the  system  will  be  in  equilibrio,  in  the  first  case,  when 

5;[P(.'r  cos  y—z  cos  «)]=0, 
and  in  the  second,  when 

5;[P(y  cos  y—z  cos  /3)]=:0. 

135.  When  the  body  is  capable  of  shding  along  the  fixed 


FORCES    ACTING    IN   SP^ACE.  69 

axis,  supposed  to  be  that  of  z,  an  additional  condition  of 
equilibrium  becomes  necessary  ;  this  condition  is  expressed 
by  the  equation 

2:(Pcos  y)=0. 

136.  By  comparing  the  conditions  of  equilibrium  of  a  sys- 
tem moveable  about  a  fixed  axis,  with  those  which  obtain 
when  the  system  turns  about  a  fixed  point,  we  iiifer.  That 
an  eqniUhriiirii  icill  take  place  about  tJie  fixed  point  tvlien,  by 
vegarding  the  axes  passing  through  this  point  as  fixed  in 
succession,  the  equilibrium  is  tnaintained  with  reference  to 
each  of  them. 

137.  If  the  forces  be  supposed  to  act  against  a  fixed  plane,, 
which  may  be  assumed  as  the  plane  of  x,  y,  the  components 
perpendicular  to  it  will  be  destroyed  by  the  reaction  of  the 
plane,  and  the  conditions  of  equilibrium  will  thus  be  reduced 
to  those  of  forces  acting  in  a  plane  ;  we  consequently  have 

^(P  cos  «)— 0, 
5:(P  cos  iS)=0, 
2[P(y  cos  »—x  cos  /3)]=0. 

138.  If  a  body  be  supposed  placed  on  a  fixed  plane,  being 
at  the  same  time  liable  to  be  overturned  by  the  action  of  thei 
forces  exerted  upon  it,  we  must  add  to  these  three  equations 
the  condition,  that  the  résultant  of  the  perpendicular  forces 
shall  pass  through  a  point  in  which  the  body  touches  the 
plane,  or  that  it  shall  intersect  the  plane  within  the  polygon 
formed  by  connecting  the  points  of  contact. 

139.  The  discussion  of  this  subject  will  be  terminated  by 
the  solution  of  the  following  problem  :  To  find  the  analyti- 
cal condition  expressive  of  the  existence  of  a  single  result^ 
ant  of  any  number  of  forces  situated  in  space.  The  system 
will  admit  of  a  single  resultant,  when  the  resultant  of  the 
components  parallel  to  the  axis  of  z  intersects  the  plane  of 
X,  y,  in  a  point  situated  on  the  resultant  of  the  forces  lying 
in  that  plane.  To  express  this  condition,  we  remark,  that  in 
case  of  an  equilibrium,  the  following  relations  must  subsist 
between  the  forces  parallel  to  the  axis  of  z  (Art.  128)  : 

P  cos  y-}-P'  cos  v'+P"  cos  y"-]-&:c,=Q. 


70  STATICS. 

P(:r  cos  y— Z  COS  «)-f-P'(a;'  cos  y'  —  z'   COS  «') 

+P"(a;"  COS  y"-z"  COS  «")+<fcc.=0. 

P(y  COS  y— 2;  COS  /3)4-P'(2/'  COS  y'  — 2;'  COS  /S} 

+P"  {y"  COS  y"— 5;"  COS  ô")+<fcc.=0. 
If  we  consider  P  cos  y,  the  first  of  these  forces,  as  equal 
and  directly  opposed  to  the  resultant  Z  of  all  the  others,  we 
shall  have  Z= — P  cos  y,  and 

— P  cos  y=P'  cos  y'+P"  cos  y"+&.C, 

— P(ar  cos  y—Z  COS  «)=P'(a:'  cos  y — z'  COS  «') 

+  P"(a:"  COS  y"-z"  COS  /)+«S6C. 
P(2/  COS  y — 2;  C0S;S)  =  P'(2/'  COS  y' — 2;'  COS/3') 

+P"(a:"  COS  y"— ;3"  cos^")+(fcc. 
The  point  of  application  of  the  resultant  being  supposed  in 
the  plane  of  x^  y,  let  x;,  y^,  and  0  be  the  co-ordinates  of  this 
point  ;  these  values,  being  substituted  in  the  first  members  of 
the  preceding  equations,  give 

— P  cos  y  3=  P'  COSy'-fP"  COS  y"-f  tfec, 
— P  COS  y.r,  =  P'(x'  COS  y'~Z   COS  a') 
-\-V'{x'  COS  y" — Z''  COS  «")  +  &c., 
P'  C0Syy,  =  P'(3/'  COS  y'—z'  COS  S') 

4-P"(y"  COS  y"—z"  cos^'')4-<fcc.  ; 
and  denoting  by  M  and  N  the  second  members  of  the  two 
last  equations,  and  replacing  the  factor  — P  cos  y  by  its  value 
Z,  we  obtain 

Z=P'cosy'-f&c., 

Zy;=N; 
whence  we  deduce 

_M        _N 
^~T    ^~Z' 

Having  thus  obtained  the  values  of  the  co-ordinates  of  the 
point  at  which  the  resultant  of  the  parallel  forces  intersects 
the  plane  of  x,  y,  it  remains  to  express  the  condition  that  this 
point  shall  be  found  on  the  direction  of  the  resultant  of  those 
forces  which  are  situated  in  the  plane  of  .r,  y  ;  the  equation 
of  the  latter  resultant  (Art.  Ill)  is 


THEORY    OF   THE    PRINCIPAL    PLANE.  71 

Xy— Yx=:s[P(y  cos  x-  x  cos  /s)]  ; 
and  putting-,  for  brevity, 

j:[P(y  cos  K—x  cos  P)]=L, 
it  becomes 

Xy— Yx=L; 
replacing  x  and  y  in  this  equation  by  the  values  of  x^,  and  y„ 
determined  above,  the  required  condition  will  be  expressed, 
and  we  shall  obtain 

XN     YM    T 

or,  by  reduction, 

XN=^LZ+MY (68). 

If  this  equation  be  satisfied,  the  system  will  admit  of  a  single 
resultant,  except  in  the  case  when 

X=0,     Y=0,     Z=0. 

140.  When  the  forces  are  situated  in  the  same  plane,  the 
system  will  in  general  admit  of  a  single  resultant  ;  for  the 
quantities  M  and  N  which  represent  the  sums  of  the  moments 
taken  with  reference  to  the  planes  of  x,  z,  and  y,  z,  being 
equal  to  zero,  as  also  the  quantity  Z  which  expresses  the 
sum  of  the  components  P'  cos  y',  P"  cos  y",  &,c.,  the  equation 
(68)  will  be  satisfied. 

141.  It  appears  from  Art.  114  that  the  equations  X=0 
and  Y=0  express  the  condition  that  the  forces  lying  in  the 
plane  of  x,  y  may  be  reduced  to  two  equal  resultants  R'  and 
R",  parallel  to  each  other,  and  acting  in  contrary  directions. 
By  a  similar  process,  the  forces  parallel  to  the  axis  of  z  may 
be  reduced  to  two,  Z'  and  Z",  equal  and  acting  in  contrary 
directions.  Hence,  when  we  have  simply  the  conditions 
X=0,  Y=0,  and  Z=0,  the  system  maybe  reduced  to  four 
forces  R',  R",  Z',  Z".     These  may  be  still  further  reduced  to 

two  equal  forces,  having  parallel  and  contrary  directions. 

Theory  of  the  principal  Plane,  and  Analogy  existing  be- 
tioeen  Projections  and  Moments. 

142.  The  theory  of  the  principal  plane,  which  presents 
results  so  nearly  allied  to  those  obtained  in  the  theory  ofino- 


72  STATÎCS. 

merits,  is  of  such  importance  m  the  higher  branches  of  me^ 
chanics,  as  to  forbid  its  omission  in  an  elementary  treatise. 
It  is  founded  on  a  theorem  demonstrated  in  the  elementary 
treatises  on  the  Differential  Calculus,  which  may  be  enun- 
ciated as  follows  :  The  projectmi  of  a  plane  surface  upon  a 
plane  is  equal  to  the  area  of  this  surface  multiplied  by  the 
cosine  of  the  angle  of  inclination. 

It  follows,  from  this  theorem,  that  if  <p  represent  the  angle 
formed  by  two  planes,  and  a  the  area  of  a  surface  situated  in 
the  first  plane,  the  projection  of  this  area  on  the  second  plane 
will  be  expressed  by  x  cos  <p.  But  the  angle  ç  included  be- 
tween the  two  planes  MF  and  EN  {Pig.  68)  is  equal  to  that 
included  between  the  two  perpendiculars  demitted  from  a 
point  C  on  these  planes.  If  one  of  these  planes,  EN  for  ex- 
ample, be  supposed  that  of  x,  y,  the  perpendicular  All  will 
become  parallel  to  the  axis  of  z.  Thus  the  angle  formed  by 
the  plane  MF  with  that  of  x,  2/,is  measured  by  the  angle  in- 
cluded between  the  perpendicular  BK  and  the  line  AH  par- 
allel to  the  axis  of  z. 

In  general,  if  «,  /3,  and  y  represent  the  angles  formed  by 
the  perpendicular  to  a  given  plane  with  the  three  co-ordinate 
axes  of  a:,  y,  and  ;r,  these  angles  will  measure  the  inclinations 
of  the  assumed  plane  to  the  planes  of  y,  2;,  .r,  z,  and  a-,  y,  re- 
spectively. 

143.  Let  a,  |3,  y,  and  f,  «',  .-",  represent  the  angles  formed 
respectively  by  any  two  planes  with  the  three  co-ordinate 
planes,  these  angles  being  equal  to  those  formed  by  the  per- 
pendiculars to  the  given  planes  with  the  axes  of  co-ordinates. 
By  introducing  the  cosines  of  these  angles  in  the  formula 
expressing  the  value  of  the  cosine  of  the  angle  included  be- 
tween two  lines,  the  value  of  their  inclination  ç>  may  he 
determined. 

If  we  draw  through  the  point  C  {Pig.  69)  the  lines  CA 
and  CB  perpendicular  to  the  given  planes,  these  lines  will 
contain  between  them  the  angle  ^,  and  its  value  will  result 
from  the  formula 

cos  p=cos£  cos  «-f  cos  t  cos /î  4- COS  î"  cos  y (71). 

144.  When  the  angle  ^  is  a  right  angle,  its  cosine  will  be 
equal  to  zero,  and  the  equation  becomes 


THEORY    OF   THE    PRINCIPAL    PLANE.  73 

COS  t  COS  «-fcOS  s  COS  jS-fCOS  t"  COS  y  =  0. 

145,  From  the  formula  (71)  we  deduce  a  very  remarkable 
property  of  projections.  For,  let  there  be  two  planes,  the  first 
of  which  forms  with  the  co-ordinate  planes  the  angles  a,  6, 
and  c,  and  the  second  the  angles  «,  /3,  y  ;  the  angle  <p  included 
between  these  planes  being  deduced  from  the  formula  (71), 
we  have 

cos  ^=cos  a  cos  <«  +  cos  h  cos  /S+cos  c  cos  y. 
But  if  we  represent  by  x  the  area  of  a  plane  surface  situated 
in  the  first  plane,  the  preceding  equation  being  multiplied 
by  A,  gives 

A  cos  <p=x  cos  a  cos  «+ a  cos  h  cos  /3+a  cos  c  cos  y (72). 

The  product  a  cos  ç>  is  equal  (Art.  142)  to  the  projection  of  the 
area  x  on  the  second  plane,  and  the  products  x  cos  a,  x  cos  6, 
and  X  cos  c  are,  in  like  manner,  the  projections  of  the  same 
area  on  the  co-ordinate  planes. 

146.  The  equation  (72)  therefore  gives  rise  to  the  following 
theorem  :  The  j)rojection  of  a  plane  surface  on  any  plane  is 
equal  to  the  simi  of  the  jiroducts  of  its  projections  on  each  of 
the  co-ordinate  planes,  nudtiplied  respectively  by  the  cosines 
of  the  angles  o,  (S,  and  y,  which  measure  the  inclinations  of 
the  plane  of  projection  to  the  co-ordinate  planes. 

This  theorem  becomes  much  more  general,  if,  instead  of  the 
area  x  lying  in  a  single  plane,  we  consider  several  areas 
X,  a',  a",  «fee.  situated  in  different  planes,  and  projected  on  a 
plane  whose  inclinations  to  the  co-ordinate  planes  are  de- 
noted by  u,  j3,  and  y  :  to  avoid  repetition,  let  us  call  the  plane 
of  projection  x,  /3,  y,  and  denote  by 

ç  and  )  the  inclinations  of  the  (  to  ttie  plane  «,  /j,  y,  and 
a,  b,  c,  )  area  a  (  to  the  co-ordinate  planes, 

the  inclinations  of  the  \  to  the  plane  «,  ,3,  y,  and 
area  a'  (  to  the  co-ordinate  planes. 

ç"  and  }  the  inclinations  of  the  C  to  the  plane  u,  /s,  y,  and 
a",  b",  c",  )  area  a"  I  to  the  co-ordinate  planes, 

<fcc.  &C.  &.C. 

By  a  method  similar  to  that  in  which  equation  (72)  was 
obtained,  we  can  obtain  similar  expressions  for  the  projections 
of  the  different  areas  ;  thus,       ^ 


<p'  and  ) 
a',  b',  c',  S 


74  STATICS. 

A  COS  ^=A  COS  a  COS  a  +  A  COS  h   COS  /3  +  A  COS  C  COS  y, 

a'  cos  ^'=A'  COS  o!  COS  a  +  A'  COS  i'  COS  /S  +  A'  COS  c'  COS  y, 

a"  COS  ç"  =>^'  COS  a"  COS  <*  +  /'  cos  6"  cos  /S  +  a"  cos  c"  cos  y, 
(fcc.  (fcc.  &.C.  <fcc.  ; 

whence,  by  addition, 

A  COS(p+A'  C0S<8'+A"  COS^"  +  (fcc.  'l 

=(a  COS  a  +  A'  COS  a'  +x"  cos  a"4-&c.)cosrt    I  _ 

+  (a  cos  b+x'  cos  6'  + A"  cos  è"  +  <fcc,)cos  ^    T ^     '^' 

+  (a  cos  c  +  a'  cos  c'  +  x"  cos  c"+&-c.)cos  y  J 

The  first  member  of  this  equation  is  the  sum  of  the  pro- 
jections of  the  areas  a,  x',  x",  <fcc.  on  the  plane  a,  0,y]  and  the 
terms  included  within  the  brackets  express  the  sums  of  the 
projections  of  the  same  areas  on  the  co-ordinate  planes.  We 
therefore  conclude  that  the  enunciation  of  the  theorem  in 
Art.  146  will,  in  the  present  case,  require  to  be  so  modified, 
that  we  may  substitute  in  the  place  of  the  plane  area  ?.,  a  sur- 
face composed  of  any  number  of  plane  areas  x,  x',  x",  éôc. 
situated  in  different  planes  :  this  modification  renders  the 
theorem  much  more  general. 

147.  For  the  purpose  of  simplifying  the  last  equation,  let 
us  denote  by  P  the  sum  of  the  projections  of  the  areas 
\,  \',  x'',  &c.  on  the  plane  «,  /3,  y,  and  by  A,  B,  and  C  the  re- 
spective sums  of  the  projections  of  the  same  areas  on  the  three 
co-ordinate  planes  ;  the  equation  will  thus  be  reduced  to 

P=A  cos<t-fB  C0S/3  +  C  cos  y (74) 

148.  It  should  be  observed,  in  taking  the  sums  of  these  pro- 
jections, that  the  cosines  of  the  angles  which  enter  into  the 
expressions  are  positive  or  negative,  according  to  the  values 
of  a,  b,  c,  a',  b',  c',  &c  ;  thus,  these  sums  will  occasionally  be 
changed  into  differences.  For  this  reason,  we  should  under- 
stand the  enunciation  of  the  general  theorem  as  being  appli- 
cable to  the  algebraic  sums  of  the  projections. 

149.  Let  the  areas  ?>,  >,',  x",  &c.  be  now  projected  on  two 
other  planes  which  form  with  the  co-ordinate  planes  the 
angles  «',  /3',  y',  a.",  li",  y"  ;  and  denote  by  P'  and  P"  the  sums 
of  the  projections  of  x,  x',  x",  <fcc.  on  the  planes  «',  ii\  y', 
«",  ft  y",  respectively  ;  we  shall  obtain  equations  similar  to 


THEORY  OP   THE    PRINCIPAL    PLANE.  75 

(74),  and  if  we  represent,  as  above,  by  A,  B,  and  C,  the  sums 
of  the  projections  of  k,  k',  \",  &c.  on  the  co-ordinate  planes, 
we  shall  have 

P  =Acos«  +Bcosi8  -fCcosy    ^ 

F  =  A  cos  *'  4-B  cos  /3'  -f-C  cos  y'   > (75). 

F  =A  cos  a"+B  cos  0"-\-C  cos  y"  } 

150.  If  the  planes  upon  which  the  projections  P,  P',  and 
P"  are  made  be  supposed  rectangular,  their  intersections 
will  be  perpendicular  to  each  other,  and  may  therefore  be 
regarded  as  three  rectangular  axes,  which  intersect  at  a 
point  O  ;  consequently,  by  representing  these  new  axes  by  Ox' 
Op',  and  Oz',  they  will  be  respectively  perpendicular  to  the 
new  planes  of  co-ordinates  ;  but  the  axes  of  x,  y.  and  z  were 
likewise  perpendicular  to  the  primitive  co-ordinate  planes  ; 
hence,  the  angles  formed  by  the  primitive  axes  with  the  new, 
will  be  measured  by  the  inclinations  of  the  primitive  co- 
ordinate planes  to  the  new.  These  angles  of  inclination  are, 
by  hypothesis  *,  /?,  y  ;  *',  ^\  y  ;  «",  /2",  y"  5  and  since  each  of 
the  primitive  axes  corresponds  to  the  same  letter  although 
differently  accented,  we  find  that 

The  axis  of  x  forms  with  the  new  axes  the  angles  «,  «',  <»", 
The  axis  of  y  forms  with  the  new  axes  the  angles  /;,  /3',  /3", 
The  axis  of  z  forms  with  the  new  axes  the  angles  y,  y',  y". 

The  following  relations  will  therefore  subsist  between  the 

cosines  of  these  angles, 

COS^  « -{-cos''   *'-|-C0S2  a"=l    ^ 

C0S2  /8  -fcos*  yfi'-f  cos2  ^"=1  > (76). 

COS^   y-4-COS'  y'-fC0S2  y"  =  l   ^ 

Again,  since  the  angle  formed  by  any  two  of  the  primitive 
axes  is  a  right  angle,  we  shall  obtain  (Art.  144) 

cos  *  cos  yg-fCOS  *'  COS  i^'-j-cos  a'  COS  (S"=0  ^ 

cos  tf  COS  y-l-COS  «'  COS  y'-f  COS  a'  COS  y"=0  > ^'l^)' 

COS  ^  COS  y-f-COS  /j'  COS  y'-f  COS  ^"  COS  y"=0  J 

151.  If  we  take  the  sum  of  the  squares  of  the  equations 
(75),  reducing  by  means  of  (76)  and  i^l)^  we  shall  obtain  the 
relation 

P»  4.F3  -j-prrs  ^p^i  -}-B^  -f  C^ (78)  ; 


76  STATICS. 

which  expresses  that  the  sum  of  the  squares  of  the  projec- 
tions of  the  areas  a,  a',  a",  <fec.  on  any  three  rectangular  planes 
is  a  constant  quantity. 

152.  Several  important  consequences  may  be  deduced 
from  this  theorem  :  thus,  if  we  resolve  the  equation  (78)  with 
reference  to  P,  we  find 

P=^(A^  H-B^»  +C*  — ?'==  — P"2). 
The  value  of  P  will  evidently  be  greatest  when  P'  and  P''' 
are  equal  to  zero.     In  this  case,  the  sum  of  the  projections  of 
A,  a',  a",  &.C.  on  the  plane  «,  /3,  -y,  will  be  given  by  the  equation 

P-v^(A2+B2+C^) (79). 

But  the  angles  t,  a',  a",  being  the  angles  formed  by  the  primi- 
tive axis  of  a;,  with  the  three  new  axes,  we  must  have  the 
relation 

A=PC0S  «+ P'  cos  a'+P"  cos  cl"  ] 

and  by  considering  the  other  angles,  we  obtain  in  Hke  msuiner, 

B=P  cos  /3+P'  cos  /S'+P  "  cos  B", 

C=P  cos  î'+P'  cosy+P"  cos/'. 

If  we  suppose,  as  above,  the  quantities  P'  and  P"  to  be  equal 

to  zero,  the  preceding  equations  reduce  to 

A=P  cos  a,    B=P  cos  0,     C=P  cos  V (80). 

whence, 

A  ^    B  C 

COSce  =  — -,      C0S/3  =  — ,      008?=—, 

and  by  substituting  for  P  its  value  given  in  equation  (79)^ 
we  find 

A 

B 

^^'''"^/(A^+B^+C^) 

C 

cos>-^^^2_|_Bs_j_C3) 

These  angles  express  the  inclinations  of  the  plane  of  maxi- 
mum projections^  which  is  called  the  principal  plane. 

The  determination  of  this  plane  being  dependent  only  on 
the  angles  «,  0,  y,  the  same  property  will  be  enjoyed  by  every 
parallel  plane. 


(81). 


THEORY   OP   THE    PRINCIPAL    PLANE.  77 

153.  It  may  also  be  demonstrated  that  the  sum  of  the  pro- 
jections of  the  areas  a,  a',  V,  (fee,  on  every  plane  which  is 
equally  inclined  to  the  principal  plane,  will  be  equal  to  a  con- 
stant quantity.  For,  let  Q,  be  the  sum  of  the  projections  on 
any  plane  whose  inclinations  to  the  co-ordinate  planes  are 
denoted  by  a,  6,  and  c  :  if  we  represent,  as  heretofore,  by  A, 
B,  C  the  projections  of  these  areas  on  the  co-ordinate  planes, 
we  shall  have 

Cl=A  cos  a-f  B  cos  b  +  C  cos  c  ; 
but  if  d,  /3,  y  denote  the  inclinations  of  the  principal  plane, 
the  equations  (80),  which  are 

A=Pcos'',     B=Pcos/3,     C=Pcosy, 
will  reduce  the  preceding  equation  to 

Q=P(cos  a  cos  *-fcos  b  cos  /3-1-cos  c  cos  y). 

The  quantity  within  the  brackets  being  equal  to  the  cosine 
of  the  angle  included  between  the  principal  plane  «,  (3,  y  and 
the  assumed  plane  a,  b,  c,  we  shall  have,  by  calling  this  in- 
clination 6, 

Q,— P  cos  6; 
and  since  P  represents  the  sum  of  the  projections  on  the  prin- 
cipal plane,  which,  by  Art.  152,  is  equal  to  .y/iA'  -f  B"  +0^), 
the  substitution  of  this  value  gives 

a=v/(A2-fB='-fC2)Xcos<» (82). 

But  the  projections  A,  B,  and  C  remaining  the  same,  it  follows 
from  the  equation  (82)  that  the  value  of  Q,,  the  sum  of  the 
projections  on  any  plane,  will  be  constantly  the  same  for  all 
planes  having  the  same  inclination  to  the  principal  plane. 

It  also  appears  that  this  sum  will  increase  or  diminish  in 
the  same  ratio  as  cos  $. 

154.  Lastly,  it  may  be  remarked  that  the  sum  of  the  pro- 
jections on  every  plane  perpendicular  to  the  principal  plane 
is  equal  to  zero  ;  for  6=90°  gives  cos  fl=0,  and  Q,=0. 

155.  The  several  theorems  relative  to  projections  which 
have  just  been  demonstrated  are  likewise  applicable  to  the 
case  of  moments.  For,  let  the  centre  of  moments  be  sup- 
posed to  coincide  with  the  origin  of  co-ordinates,  and  con- 
ceive the  plane  «,  0,  y  to  pass  through  the  origin  :  if  from  the 


78  STATICS. 

points  of  application  of  the  several  forces  we  take  upon  their 
respective  hnes  of  direction,  portions  wliich  shall  be  propor- 
tional to  the  intensities  of  these  forces,  these  lines  may  be 
represented  by  the  letters  P,  P',  P",  &.c.  The  centre  of  mo- 
ments may  then  be  regarded  as  the  common  vertex  of  several 
triangles,  of  which  P,  P',  P",  (fee.  represent  the  bases  :  the 
projections  of  these  triangles  upon  the  plane  «,  /3,  y,  and  on 
the  co-ordinate  planes  will  likewise  be  triangles,  their  bases 
2^,  ])',  p",  (fee,  being  the  projections  of  the  lines  P,  P',  P",  (fee, 
and  their  altitudes  h,  A',  h",  (fee,  being  the  perpendiculars  de- 
mitted  on  the  lines  p,  p',  p",  (fee.  from  the  centre  of  moments. 
These  values  behig  substituted  in  eq^nation  (73),  which 
may  be  written  under  the  following  form  : 

2(the  projections  on  the  plane  «,  /3,  y)  = 
{  The  projections  on  the  co-ordinate  planes  multipHed  ) 
(  respectively  by  the  cosines  of  the  angles  of  inclination  y 

convert  it  into 

Iph  +  \p'h' + \p"h!'  -1-  (fee.  = 
(  The  projections  on  the  co-ordinate  ^ 

s  ^  planes  multiplied  respectively  by  the  j> (83).. 

1^  cosines  of  the  angles  of  inclination  J 

The  second  member  of  t'his  equation  will  contain  similar  pro- 
ducts, and  the  factor  \  will  therefore  be  common  to  the  two 
members  ;  this  being  suppressed,  the  first  member  will  re- 
duce to 

ph-{-p'h'^p"k!'^6i.c. 
But  p,  p',  p",  (fee,  being  the  projections  of  the  right  lines  P, 
P',  P",  (fee,  the  products  j^h,  p'h',  p"h",  tfee  will  be  the  mo- 
ments of  the  lines  />,  p',  jy",  (fee,  taken  with  reference  to  the 
origin  of  co-ordinates.  The  same  remarks  being  applicable 
to  the  second  member  of  equation  (83),  it  follows  that  the 
sum  of  the  moments  of  the  projections  of  the  forces  on  the 
plane  «,  /3,  y,  which  passes  through  the  origin  of  co-ordinates, 
is  equal  to  the  sum  of  the  moments  of  the  projections  of  the 
same  forces  on  tlie  three  co-ordinate  planes,  multiplied  re- 
spectively by  the  cosines  of  the  angles  of  inclination. 

156.  By  making  similar  substitutions  in  equations  (78),  it 
may  likewise  be  proved  that  the  sum  of  the  squares  of  the 


CENTRE    OF    GRAVITY.  7*^ 

moments  of  the  different  forces,  when  projected  on  three 
rectangular  planes,  is  a  constant  quantity. 

The  equations  (80)  make  known  the  position  of  the  plane 
in  which  the  sum  of  the  moments  will  be  the  greatest  pos- 
sible. And  the  equatian  (79)  determines  the  sum  of  the 
moments  on  the  principal  plane. 

Centre  of  Gravity. 

157.  The  particles  of  matter  are  constantly  subjected  to  the 
action  of  a  force  which  tends  to  draw  them  towards  the  earthy 
in  directions  perpendicular  to  its  surface.  This  force  is  called 
the  force  of  gravity. 

The  earth  being  nearly  spherical,  the  lines  of  direction  in 
which  material  points  tend  to  move,  will  converge  towards 
its  centre  ;  and  since  the  distance  of  this  centre  from  the  sur- 
face is  exceedingly  great  when  compared  with  the  dimen- 
sions of  those  objects  which  we  usually  consider,  the  direc^ 
tions  of  the  forces  which  act  on  the  different  particles  of  the 
same  body^  may,  without  sensible  error,  be  regarded  as 
parallel. 

158.  It  is  known  from  observation  that,  as  we  recede  from 
the  centre  of  the  earth,  the  intensity  of  gravity  diminishes  in 
the  inverse  ratio  of  the  square  of  the  distance  included  be- 
tween the  centre  and  the  place  of  observation.  For  example, 
if  a  body  be  placed  at  a  certain  distance  from  the  centre  of  the 
earth,  assumed  as  unity,  and  be  subsequently  transported  to 
distances  represented  by  2,  3,  4,  &c.,  the  intensity  of  the  force 

of  gravity  will  become  _,_,_,  &c.,  or  -^  ^,  --,  &c.,  of 

what  it  was  at  the  distance  of  unity. 

159.  The  earth  being  flattened  towards  the  poles,  and  pro- 
tuberant at  the  equator,  it  follows,  that  in  going  from  the 
equator  towards  the  poles,  we  must  necessarily  approach  the 
centre  of  the  earth,  and  the  intensity  of  gravity  will  therefore 
increase.  It  will  appear  hereafter  in  discussing  the  subject  of 
centrifugal  forces,  that  from  another  cause,  the  intensity  of 
the  force  of  gravity  is  greater  at  the  poles  than  at  all  other 
places  on  the  earth's  surface. 


80 


STATICS. 


160.  The  action  of  gravity  being-  exerted  on  all  the  particles 
which  compose  a  body,  these  particles  may  be  regarded  as 
solicited  by  forces  whose  directions  are  parallel  ;  the  resultant 
of  these  forces  is  equal  to  their  sum,  and  constitutes  what  is 
called  the  weight  of  a  body.  Hence,  if  the  bodies  considered 
are  homogeneous  with  each  other,  their  weights  will  be  pro- 
portional to  their  volumes. 

161.  The  term  density  is  used  to  express  the  gi*eater  or 
less  number  of  particles  contained  in  a  body  of  a  given 
volume,  when  compared  with  the  number  of  particles  con- 
tained in  some  other  body  assumed  as  a  standard.  If  we 
assume  as  the  unit,  the  quantity  of  matter  contained  in  a 
cubic  foot  of  a  given  substance,  distilled  water  for  example, 
and  compare  this  quantity  with  that  contained  in  a  cubic 
foot  of  any  other  substance,  their  ratio  will  express  the  density 
of  the  second  substance.  Let  this  ratio  be  denoted  by  D. 
If  the  second  substance  considered  were  gold,  by  calling  D 
the  density  of  gold,  we  should  have 

The  quantity  of  matter  in  )  _  (  D  X  T/ie  quantity  of  matter 
a  cubic  foot  of  gold  )       (      in  a  cubic  foot  of  water  ; 

whence 

y^_  (Quantity  of  'matter  in  a  cubic  foot  of  gold 
Quantity  of  matter  in  a  cubic  foot  of  water 

162.  In  the  preceding  article  we  have  considered  bodies  of 
the  same  volume  ;  but  if  we  wish  to  estimate  the  quantity  of 
matter  contained  in  a  homogeneous  body  whose  volume  is  V, 
the  quantity  D  must  be  taken  as  many  times  as  there  are 
units  of  volume  in  the  volume  V  ;  we  shall  thus  have 

M=DV (84). 

The  quantity  M  is  called  the  mass,  and  evidently  expresses 
the  relation  between  the  quantity  of  matter  contained  in  the 
body,  and  that  contained  in  the  unit  of  volume  of  the  sub- 
stance assumed  as  the  standard. 

163.  If  the  intensity  of  gravity  were  the  same  at  all  places, 
the  weight  of  a  body  would  be  proportional  to  its  mass,  and 
might  be  represented  by  the  same  quantity.  For,  if  ^^  denote 
the  effect  exerted  by  gravity  on  the  unit  of  mass,  or  the 
weight  of  the  unit  of  mass,  and  W  the  weight  of  the  body,  we 


CENTRE   OP   GRAVITY.  81 

shall  have,  from  the  definition  of  the  weight,  W=M^  ;  in 
which  expression  the  quantity  g  will  be  constant,  and  may 
be  assumed  as  the  unit  ;  we  shall  thus  obtain  the  relation 

W=M (85). 

This  equation  merely  expresses  that  the  number  of  units  of 
weight  is  equal  to  the  number  of  units  of  mass. 

But,  if  by  transporting  the  mass  to  different  distances  from 
the  earth's  centre,  the  intensity  of  gravity  be  subject  to  varia- 
tion, the  quantity  g  will  be  variable,  and  the  equation  ex- 
pressing the  relation  between  the  mass,  weight,  and  intensity 
of  gravity,  must  then  be  written  under  the  general  form 
W=M^ (86). 

164.  From  the  equations  (84)  and  (86),  we  deduce 

which  indicates  that  the  weight  varies  proportiorially  to  the 
gravity-  g,  the  volmne  V,  and  the  density  D. 

165.  If,  for  example,  two  bodies  of  the  same  volume  be 
subjected  to  the  action  of  the  same  force  of  gravity,  their 
weights  will  be  in  the  direct  ratio  of  their  densities. 

The  intensity  of  gravity  varying  only  with  change  of 
place,  it  follows  that^  will  be  constant  for  all  bodies  at  the 
same  place. 

166.  If  there  be  any  number  of  points  firmly  connected 
together,  and  solicited  by  the  weights  P,  P',  P",  &c.,  we  may 
regard  these  weights  as  parallel  forces  ;  and  denoting  the 
co-ordinates  of  the  respective  points  by  x,  y,  z,  a/,  y',  z\ 
x",y",  z"j  6cc.,we  shall  obtain,  from  Art.  (80)  and  (81),"  the 
expressions  for  the  co-ordinates  of  the  centre  of  parallel  forces  ; 
these  co-ordinates  being  represented  by  X/,  yi,  Zj,  we  find 

'Px  +  F'x'  +  V"x"  +  ôcc. 


x,= 


P-fP'  +  P"+<fcc.  ' 
Py-fPy  +  P^y^  +  &c. 

P  +  P'  +  P"+&c.  ' 
Pz-i-P'z'+P"z"+&c. 


P+P'+P"  +  &c. 
167.  When  the  forces  are  exerted,  as  in  the  present  in- 
stance,  by  the  action  of  gravity,  the  centre  of  parallel  forces 
is  called  the  centre  of  gravity.    Let  m,  ml\  m",  (fcc.  represent 

F 


WS  STATICS. 

the  masses  corresponding  to  the  weights  P,  P',  P",  &c.,  we 
shall  have 

P=mg,    P'  ='m'g;    P"  =  m"g,  <fec.  ; 
and  by  substituting  these  values  in  the  preceding  equations, 
omitting  the  factor  g-,  which  is  common  to  the  numerators 
and  denominators  of  the  fractions,  we  obtain 

mx^'m'.T'-^m"x"  +  &^c. 


m-^-m'-^-ni" +ÔÙC.     ' 

_my-^on'y^-\-Qn"y"  -{-éùc. 

'm-^m' +?n"-\-âcc.     ' 

_mz-\-'m'z'-\-m"z"-{-&:-c.  ^ 

■m-j-ni'  -i-m" +ÔCC.      ' 

whence  it  appears  that  the  position  of  the  centre  of  gravity 

is  independent  of  the  intensity  of  the  force  of  gravity. 

168.  If  the  bodies  are  composed  of  a  homogeneous  sub- 
stance, the  density  of  which  is  represented  by  D,  we  shall 
have,  by  denoting  their  volumes  by  v,  v',  v",  ôcc.  (Art.  162), 

m=vD,    m'=v'D,    m"  =v"T>,  <Scc.  ; 
and  by  a  substitution  and  reduction  similar  to  the  preceding, 
we  find 

vx-\-v'x' +v"x"-{-&c. 


xr- 


Zi  — 


v-\-v'-{-v"-\-6lc. 
vy + v'y' + v"y"  -f-  &c. 

v-\-v'-\-v"-\-&,c. 
vz + v'z'  -f-  v"z" + <fec. 


or  calling  V  the  volume  of  the  entire  system,  these  equations 
become 

vx-\-v'x'-\-  v"x" + <fec. 


x,- 


yi 


_vy-\-v'y'-\-v"y"+ècc. 


vz  4-  v'z' + v"z" + &c. 
Z/= ^ 

169.  To  determine  the  centre  of  gravity  experimentally, 
we  suspend  the  body  by  a  thread  CA  {Fig.  70),  and  the  pro- 
longation AB  of  the  direction  of  this  thread  wiU  necessarily 


CENTRE    OP    GRAVITY.  83 

pass  through  the  centre  of  gravity.  The  point  in  the  Une 
AB  at  which  the  centre  of  gravity  is  situated,  may  then  be 
found  by  suspending  the  body  from  a  second  point  E  ;  the 
vertical  line  EF,  passing  through  this  point,  must  likewise 
pass  through  the  centre  of  gravity,  which  will  consequently 
be  found  at  the  point  G,  the  intersection  of  the  two  lines  AB 
and  EF. 

In  this  experiment,  the  body  is  sustained  by  that  point  to 
which  the  thread  is  attached  :  the  resultant  of  all  the  actions 
of  gravity  upon  the  particles  of  the  body  must  therefore  pass 
through  this  point,  and  its  direction  must  coincide  with  that 
of  the  thread. 

170.  The  centre  of  gravity  of  a  right  line  AB  {Fig.  71)  is 
situated  at  its  middle  point  C  :  for,  by  regarding  the  hne  as 
composed  of  heavy  material  points,  each  particle  m  situated 
on  one  side  of  the  point  C  will  correspond  to  a  particle  m'  on 
the  contrary  side,  and  equally  distant  from  the  same  point  : 
the  moments  m  X  Cm  and  m'  X  Cm'  are  therefore  equal  and 
have  contrary  signs.  The  same  remarks  are  applicable  to 
all  the  other  points  of  the  line  AB,  taken  by  pairs  ;  hence  it 
follows,  that  the  algebraic  sum  of  the  moments  of  all  the  par- 
ticles taken  with  reference  to  the  point  C  is  equal  to  zero  ; 
the  moment  of  the  resultant  taken  with  reference  to  the  same 
point  is  therefore  zero,  and  the  direction  of  the  resultant  must 
pass  through  the  point  C,  situated  in  the  middle  of  the 
line  AB. 

171.  The  centre  of  gravity  of  a  'parallelogram  AD  {Pig. 
72)  is  at  the  intersection  G  of  the  right  lines  EF  and  HK, 
which  bisect  the  parallel  sides. 

For,  if  we  conceive  the  particles  which  compose  the  paral- 
lelogram to  be  situated  on  lines  parallel  to  AB,  the  centres  of 
gravity  of  all  these  lines  will  be  found  on  the  hne  EF  drawn 
through  the  middle  points  E  and  F  of  the  opposite  sides  AB 
and  CD,  since  EF  will  bisect  all  these  parallels.  Hence,  the 
centre  of  gravity  of  the  entire  parallelogram  will  be  situated 
on  the  line  EF.  In  like  manner,  it  may  be  proved  that  the 
centre  of  gravity  lies  on  the  line  HK  which  bisects  the  sides 
AC  and  BD  ;  it  will  therefore  be  situated  at  the  point  G,  the 
intersection  of  the  two  lines  EF  and  HK. 

F2 


84  STATICS. 

172.  The  centre  of  gravity  G  of  the  area  of  a  triangle  ABC 
{Fig-  73)  is  found  by  drawing  a  line  CD  from  the  vertex  to 
the  middle  of  the  opposite  side.,  and  taking  a  part  DG  equal 
to  one-third  of  the  whole  line  CD.  For,  since  the  line  CD 
passes  through  the  middle  of  all  the  lines  parallel  to  the  base 
AB,  it  contains  the  centre  of  gravity  of  the  area  of  the  tri- 
angle :  for  a  similar  reason,  this  centre  must  lie  on  the  line 
AE  drawn  through  the  middle  of  the  side  CB  :  hence,  it  is 
found  at  the  point  G,  the  intersection  of  these  two  lines.  But, 
by  connecting  the  points  D  and  E,  we  form  the  triangle  BED, 
which  is  similar  to  the  triangle  BCA,  since  the  two  triangles 
have  a  common  angle,  and  the  sides  adjacent  directly  propor- 
tional :  the  line  DE  is  therefore  parallel  to  AC,  and  the  tri- 
angles ACG  and  DEG  are  likewise  similar  ;  hence, 

CG  :  GD  :  :  AC  :  DE  :  :  AB  :  BD  :  :  2  :  1  ; 
from  which  results 

CG=2GD, 
and,  consequently, 

CDorCG  +  GD=3GD, 
or, 

GD  =  iCD. 

173.  To  find  the  centre  of  gravity  of  a  triangular  pyra- 
mid,  ive  draw  through  the  vertex  and  the  centre  of  gravity 
of  tlte  base,  the  line  AG  {Pig.  74),  and  take  the  distance 
GO  =  iAG  ;  the  jmint  O  will  be  the  centre  of  gravity  of  the 
pyramid. 

For,  if  we  conceive  the  pyramid  divided  into  an  infinite 
number  of  elements  by  planes  parallel  to  the  base  BCD  ;  the 
line  AG  will  pass  through  the  centres  of  gravity  of  all  these 
elements,  and  will  therefore  contain  the  centre  of  gravity  of 
the  pyramid.  In  like  manner,  by  drawing  the  line  DG' 
through  the  vertex  D  and  the  centre  of  gravity  G'  of  the  oppo- 
site face,  this  line  will  also  contain  the  centre  of  gravity  of  the 
pyramid.  But,  since  the  lines  AG  and  DG'  are  situated  in  the 
plane  of  the  triangle  AED,  and  are  not  parallel,  they  will 
intersect,  and  hence  the  centre  of  gravity  of  the  pyramid  will 
be  found  at  O,  their  point  of  intersection. 

The  points  G  and  G'  being  connected,  the  triangles  EGG' 


CENTRE    OF    GRAVITY. 


85 


and  EDA  will  be  similar,  since  they  have  a  common  angle  E, 
and  the  sides  containing  it  directly  proportional  ;  hence,  GG' 
is  parallel  to  AD,  and  the  triangles  AOD,  GOG'  are  likewise 
similar  ;  from  these  we  obtain 

GG'  :  AD  :  :  GO  :  OA  ; 
but  the  similar  triangles  EGG'  and  EDA  give 

GG'  :  AD  :  :  EG  :  ED  ; 
whence,  by  comparing  these  two  proportions,  we  have 

GO  :  OA  :  :  EG  :  ED  :  :  1  :  3  ; 
and  from  this  proportion  we, find 

3G0=0A  ; 
adding  GO  to  each  member  of  the  equation,  there  results 

4G0=0A+G0=AG, 
or, 

GO  =  iAG. 
174.  In  general,  the  centre  of  gravity  of  any  pyramid  (Fig, 
75)  is  situated  on  the  right  line  SF,  drawn  from  the  vertex 
to  the  centre  of  gravity  of  the  base,  and  at  a  distance 
FO  =  |^SF.  Tf  we  draw  through  the  point  O  thv  ;  deter- 
mined, a  plane  parallel  to  the  base  of  the  pyramid,  this  plane 
will  contain  the  centre  of  gravity  of  the  pyramid.  For,  if 
through  F,  the  centre  of  gravity  of  the  polygonal  base,  the 
lines  FA,  FB,  »fcc.  be  drawn  to  the  several  angles  of  this  poly- 
gon, we  shall  form  as  many  triangles  as  the  figure  has  sides, 
and  these  triangles  may  be  regarded  as  the  bases  of  triangu- 
lar pyramids  having  a  common  vertex  S.  The  lines  drawn 
from  the  vertex  S  to  the  centres  of  gravity  of  the  several 
triangles  will  be  cut  proportionally  by  the  plane  parallel  to 
the  base,  and  the  points  of  intersection  will  therefore  be  situ- 
ated at  distances  from  the  base,  equal  to  one-fourth  of  the 
distance  from  the  base  to  the  vertex  of  the  pyramid.  Hence, 
these  points  of  intersection  will  be  the  centres  of  gravity  of 
the  several  triangular  pyramids.  But  the  centres  of  gravity 
of  all  the  partial  pyramids  being  situated  in  the  same  plane 
parallel  to  the  base,  it  follows  that  the  centre  of  gravity  of  the 
whole  pyramid  will  likewise  be  situated  in  this  plane.  It 
must  also  be  found  on  the  line  SF,  which  contains  the  centres 

8 


86  STATICS. 

of  gravity  of  all  the  sections  parallel  to  the  base,  and  we 
therefore  conclude  that  the  centre  of  gravity  of  any  pyramid 
is  situated  on  the  line  drawn  from  the  vertex  of  the  pyramid 
to  Hie  centre  of  gravity  of  the  base,  and  at  a  distance  from, 
the  base  equal  to  one-fourth  of  the  entire  distance  to  the  vertex. 

175.  To  find,  the  centre  of  gravity  of  the  area  of  a  polygon. 
Let  the  polygon  be  divided  into  triangles  {Fig.  76),  and  de- 
note by  a,  a',  a",  (fee,  the  areas  ABC,  ACD,  ADE,  «fee.  of 
these  triangles  :  let  weights  proportional  to  a,  a',  a",  (fee.  be 
supposed  applied  at  the  centres  of  gravity  G,  G',  G",  (fee,  of 
the  several  triangles.  The  centre  of  gravity  of  the  area 
ABCDA  may  then  be  found  by  the  proportion 

a+a'  :  a  :  :  GG'  :  G'O. 
In  like  manner,  the  centre  of  gravity  K  of  the  area  ABCDEA 
may  be  found  by  determining  the  resultant  of  a-\-a"  acting 
at  O,  and  a"  acting  at  G".     Its  position  will  be  ascertained 
by  the  proportion 

a  +  a'+«":  a"::  OG":  OK; 
and  the  same  process  may  be  continued  for  any  number  of 
triangles. 

176.  This  problem  may  also  be  solved  by  means  of  the 
equations  of  parallel  forces.  For  let  Xj  and  y,  denote  the  co- 
ordinates of  the  centre  of  gravity  of  the  polygon  {Fig.  77)  : 
from  the  theory  of  parallel  forces  we  obtain  the  equations 

R=P+F-fP"4-P"', 

'Rx,^Vx-^V'x'+V"x"+V"'x"',       * 
%^ = Py  -f  V'y' + V"y"  -^  V"'y"'. 
And  denoting  as  above  by  a,  a',  «",  a'",  the  areas  of  the  tri- 
angles ABC,  ACD,  ADE,  AEF,  we  shall  have,  since  the  areas 
may  be  substituted  for  the  weights  to  which  they  are  pro- 
portional, 

P=a,     F=a',     P"=a",     P"'=a"' ; 
and  the  preceding  equations  become 
R=a-|-a'-f-a"  +  a"', 

ax-\-a'x'^d'x"-k-a"'x"' 
^'~"        a^a:^a:'-\-a:"       ' 

ay-{-a'y'+a"y"+ al'Y' 
^'"^        a+a'+a"-^a"''     * 


CENTRE    OP   GRAVITY.  87 

Thus,  having-  taken  the  part  OP=a:,,  we  draw  the  line  PG 
parallel  to  the  axis  of  y  and  equal  to  y,  ;  the  point  G  will  be 
the  centre  of  gravity. 

177.  To  find  the  centre  of  gravity  of  the  perimeter  of  a 
polygon.  We  proceed  in  the  present  case  in  a  manner  similar 
to  that  adopted  in  the  preceding  example,  merely  observing 
that  the  centre  of  gravity  of  each  side  will  be  situated  at  its 
middle  point,  and  that  these  points  may  be  regarded  as 
loaded  with  weights  proportional  to  the  sides. 

178.  To  find  the  centre  of  gravity  of  the  arc  of  a  jylane 
curve.  If  the  curve  be  divided  into  elementary  portions, 
the  value  of  the  element  mm'  {Fig.  78)  will  be  expressed  by 
*/{dx^  -\-dy^\  and  since  this  element  is  indefinitely  small,  its 
centre  of  gravity  may  be  regarded  as  coinciding  with  its 
middle  point  o,  and  having  the  same  co-ordinates  x  and  y  as 
the  point  m  ;  the  moment  of  TnmJ  with  reference  to  the  axis 
of  :r,will  therefore  be 

op  X  tnrr^  =y  X  ^{dx^  +  dy^  ), 
and  its  moment  with  reference  to  the  axis  of  y,  will  be 

oq  X  'mm'=x  X  y'  (dx^  -\-  dy~  ). 
If  X,  and  y  I  represent  the  co-ordinates  of  the  centre  of  gravity^ 
£ind  s  the  length  of  the  curve  MM',  the  moments  of  this  arc 
supposed  concentrated  at  its  centre  of  gravity,  taken  with 
reference  to  the  axes,  will  be  respectively  sx^  and  sy^  :  and 
since  these  moments  must  be  equal  to  the  sum  of  the  mo- 
ments of  the  elements,  we  shall  have 

sx,—fx^{dx^  -\-dy^)^ 
sy,=fy^{dx^-\-dy-)\ 
and  the  length  of  the  arc  MM'  will  result  from  the  formula 
s^f^idx^^dy^). 

179.  Let  it  be  required,  for  example,  to  determine  the  centre 
of  gravity  of  the  arc  BO  of  a  circle  {Pig.  79).  The  co- 
ordinate axes  being  selected  in  such  a  manner  that  the  arc 
shall  be  bisected  by  the  axis  of  abscisses  passing  through  the 
centre  of  the  circle,  the  arc  will  be  divided  symmetrically  by 
this  axis,  and  the  centre  of  gravity  of  the  arc  will  then  be 


88 


STATICS. 


found  on  this  line;  hence,  we  shall  have  2/,=0.  It  will 
therefore  be  only  necessary  to  determine  the  absciss  AG=a:, 
of  the  centre  of  gravity  of  the  arc  BO.  But  the  value  of  x, 
results  from  Art.  178  ;  thus, 

sx,=fx^{dx''-]-dy') (87). 

To  integrate  the  second  member  of  this  equation,  we  elimi- 
nate one  of  the  variables  by  means  of  the  equation  of  the 
circle,  which  is 

y^-=a^—x^ (88); 

and  by  differentiating  this  equation,  we  obtain 

ydy^= — xdx  ; 
whence, 

X^ 

and  by  substituting  this  value  in  the  expression  '^{dx^-\-dy^\ 
we  have 

Vkdx-^-^dy-^^s/  i^^^dy^y, 

which,  reduced  by  means  of  equation  (88),  gives 

ady 
^{dx--\-dy-)=-^\ 

this  value  being  substituted  in  equation  (87),  we  find,  by 
integration 

fx^{dx^-Vdy^)=ay^^ (89), 

the  quantity  B  representing  an  arbitrary  constant. 

If  we  denote  by  c  the  chord  of  the  arc  BO,  and  wish  to 
determine  the  centre  of  gravity  of  the  arc  which  it  subtends, 
we  must  integrate  between  the  limits  2/=ic,  2/= — \c.  But 
since  the  arc  extends  from  O  to  B,  this  integral  will  become 
zero  at  the  point  O,  the  ordinate  of  which  is  y—  —  \c.  This 
supposition  reduces  equation  (89)  to 

0=  — iac+B; 
by  eliminating  B  between  this  equation  and  (89),  we  find 

fx^{dx''-\-dy'')—ay-\r\ac\ 
and  making  y=\c^  for  the  purpose  of  taking  the  entire 
integral  from  the  point  O  to  the  point  B,  we  obtain 


CENTRE   OP   GRAVITY.  89 

fxy/{dx^  •\-dy^)=ac  ; 
which  vahie  substituted  in  equation  (87),  gives 

sx,=ac, 
or 

radius  X  chord 


x.=- 


(90); 


arc 

the  absciss  of  the  centre  of  gravity  is  therefore  a  fourth  pro- 
portional to  the  arc^  the  chord,  and  the  radius. 

180.  To  find  the  centre  of  gravity  of  a  curve  of  double 
curvature,  or,  in  general,  that  of  any  line  situated  in  space. 

The  expression  for  the  element  of  a  curve  of  double  curva- 
ture being 

^{dx^-\-dy^-\-dz') (91), 

let  the  moments  of  this  element  be  taken  with  reference  to 
the  co-ordinate  planes.  The  co-ordinates  x,  y^  and  z  repre- 
sent the  distances  of  this  element  from  the  planes  of  y,  z,  x,  z^ 
and  xy,  and  the  respective  moments  will  therefore  be 

x^{dx''-\-dy'^-\-dz'') 

y^\dx^-^dy^-\-dz^)  \ (92)  ; 

z^  {dx^  ■\-dy'^  -\-dz2) 

consequently,  if  we  denote  by  x^,y„  and  z,  the  co-ordinates 
of  the  centre  of  gravity,  and  by  s  the  length  of  the  arc,  these 
quantities  will  be  determined  by  means  of  the  equations 
s-=-  f^idx^^dy^^dz^)  \ 
sx,=fx^{dx^-^dy^^dz^)   I 
sy,=fy^{dx^+dy--\-dz^)   ^  •  •  •  •  V^-^^ 
sz,=fz^{dx^-\-dy^-\-dz'^)  J 

181.  Let  it  be  required  to  apply  these  formulas  to  the 
case  of  a  right  line  situated  in  space.  Assume  the  origin  at 
one  extremity  of  the  line  ;  the  equations  of  the  line  will  then 
be  of  the  form 

x=«z,    y=fiz (94); 

whence, 

dx = ecdz,     dy  —^dz. 
These  values  substituted  in  the  expression  (91),  give 
^{dx^  -{-dy^  -\-dz^)=dz^{\+cc^  ■^^^)', 

CO    -4-    M  . 


90  STATICS. 

and  putting,  for  brevity,  the  radical  equal  to  A,  we  shall  have 

Substituting  this  value  in  the  equations  (93),  and  likewise 
those  of  X  and  y  given  by  equations  (94),  we  find 
s=fA.dz=A.Zy  • 

SX  I  =fKcczdz  =  ^Aaz^, 
syi  =fA^zdz  =  ^Aisz^; 
sz,  =fkzdz  =  1  Az  2 . 
Let  h  represent  the  ordinate  z  of  the  point  M  {Fig.  80). 
To  determine  the  centre  of  gravity  of  AM,  we  must  integrate 
between  the  limits  z=0  and  z=/t,  and  we  shall  thus  find 

s=AA, 
5a:y  =  |A«A^, 
5y;  =  |A/3/z% 

Eliminating  s^  and  reducing,  we  obtain 

^i  =  \^K     yi=ï^h,     z,  =  \h. 
These  values  correspond  to  the  co-ordinates  of  the  point  O, 
the  middle  of  the  right  line  AM  ;  for,  if  AO  be  the  half  of  AM, 
the  similar  triangles  AOQ,  AMP  will  give 

GlO  =  iMP=iA; 
which  value  being  substituted  in  equations  (94),  we  find 

x=^i>^h,    y  =  ¥^^- 

182.  To  find  the  centre  of  gravity  of  a  plane  surface,, 
bounded  by  the  arc  of  a  cvjrve^  and  the  axis  of  abscisses. 

Let  Xj  and  yj  be  the  co-ordinates  of  the  centre  of  gravity 

of  the  entire  surface,  and  let  G  be  the  centre  of  gravity  of  an 

element  MP'  {Pig.  81);  the  area  of  this  element  being  equal 

to  ydx,  its  moment  with  reference  to  the  axis  of  x  will  be 

GNxy^dx,  and  that  with  respect  to  the  axis  of  y  will  be 

ANxydx.     But  since  the  element  MP'  may  be  regarded  as 

a  rectangle  whose  side  PP'  is  mdefinitely  small,  we  shall  have 

PM 
AP=AN=a:,  and  GN=— — =^y  :  hence  the  moments  with 

reference  to  the  two  axes  become  \y^dx^  and  xydx.     If  we 
represent  by  x  the  surface  DBMP,  its  area  and  the  co-ordi- 


CENTRE   OP   GRAVITY.  91 

nates  of  its  centre  of  gravity  will  be  determined  by  means  of 
the  equations 

x=fydx,      ^ 
xxt=fxydx,    V (95'), 

183.  To  apply  these  formulas,  let  it  be  required  to  find  the 
centre  of  gravity  of  a  circular  segment  CDE  {Pig.  82).  The 
origin  being  assumed  at  the  centre  of  the  circle,  and  the  axis 
of  abscisses  AD  a  line  bisecting  the  arc  CE,  the  centre  of 
gravity  of  the  segment  will  evidently  be  situated  upon  this 
line  ;  it  will  therefore  be  only  necessary  to  calculate  the  value 
of  the  absciss  AG=X;.  If  g  and  g'  represent  the  centres  of 
gravity  of  the  semi-segments,  they  will  be  found  at  equal  dis- 
tances from  the  axis  AD,  on  a  line  gg'  perpendicular  to  this 
axis,  since  the  entire  segment  is  divided  into  two  symmetri- 
cal portions  ;  the  line  gg'  will  therefore  intersect  the  axis  of 
abscisses  at  a  point  G,  the  centre  of  gravity  of  the  entire 
segment. 

The  question  is  thus  reduced  to  determining  the  absciss  of 
the  centre  of  gravity  of  the  semi-segment  CDB,  and,  its  value 
may  be  foimd  by  integrating  the  equation  (95). 

For  the  purpose  of  eliminating  one  of  the  variables  in  this 
expression,  we  assume  the  differential  equation  of  the  circle, 

ydy—  — xdx  ; 
from  which,  by  substitution  in  equation  (95),  we  obtain 

xx^^f-y^dy (96); 

and  by  integrating,  and  introducing  a  constant  A,  we  have 

f-y^dy=-\y'+K (97). 

To  determine  the  value  of  this  constant,,  the  integral  must  be 
taken  from  the  point  C  to  the  point  D  ;  or,  if  we  denote  by 
c  the  value  of  the  chord  CE,  the  limits  of  the  integral  will  be 
2/=ic  and  2/=(X  Thus,  if  we  suppose  the  integral  to  become 
zero,  when  y=\c,  the  constant  A  will  result  from  the  equation 

and  the  equation  (97)  will  therefore  become 


Oj2  STATICS. 

Putting  y=0,  to  obtain  the  value  of  the  entire  integral  from 
C  to  D,  we  have 

This  value  substituted  in  equation  (96), gives 

""'-24^  = 
but  since  x  represents  in  this  expression  the  area  CDB,  we 
have 

A=i  area  CDEB, 
whence, 

-___£!___  • 

^'""12  area  CDEB' 
and  we  therefore  conclude,  that  the  distance  from  the  centre 
of  gravity  of  a  circular  segment  to  the  centre  of  the  circle  is 
equal  to  the  cube  of  the  chord  divided  by  twelve  times  the  area 
of  the  segment. 

184.  To  find  the  centre  of  gravity  of  a  circular  sector  CAE 
{Fig.  83).  The  centre  of  gravity  is  evidently  situated  on 
the  radius  AB  which  divides  the  sector  into  two  equal  parts  ; 
it  will  therefore  be  only  necessary  to  determine  the  value  of 
the  absciss  AG.  If  we  regard  the  sector  CAE  as  composed 
of  an  infinite  number  of  elementary  sectors,  the  centre  of 
gravity  of  each  will  be  situated  at  a  distance  from  the  point 
A  equal  to  two-thirds  of  the  radius  AC,  since  these  sectors 
may  be  considered  triangular.  Hence,  if  from  the  centre  A, 
with  a  radius  equal  to  two-thirds  of  AC,  we  describe  the  arc 
HK,  the  centres  of  gravity  of  all  the  elementary  sectors  will 
be  distributed  uniformly  along  this  arc  ;  and  consequently, 
the  centre  of  gravity  of  this  arc  will  coincide  with  that  of,  the 
circular  sector.  But  if  X)  denote  the  absciss  AG,  we  have,  by 
Art.  179, 

_ AH X chord  HK, 
*'  arc  HK        ' 

and  from  the  similarity  of  the  sectors  AHK  and  ACE,  we 
find 


CENTRE    OP    GRAVITY.  93 

AH=|AC, 
chord  HK=|  chord  CE, 
arcHK=|arc  CE; 
which  values  substituted  in  the  preceding  equation  give  by 
reduction, 

_|ACx  chord  CE 
*'  arc  CE 

185.  To  find  the  centre  of  gravity  of  an  area  OBO' 
{Pig.  84)  comprised  between  tioo  branches  of  a  curve. 

Let  y  and  y'  represent  the  two  ordinates  PM  and  PM'  cor- 
responding to  the  same  absciss  AP=x  :  the  element  MN'  of 
the  surface,  being  the  difference  of  the  areas  PN  and  PN',  will 
be  expressed  by 

ydx—y^dx—{y — y'^dx  ; 
and  if  we  represent  by  a  a  portion  of  the  area  included 
between  the  chords  MM'  and  00',  we  shall  have 

The  element  MN'  being  regarded  as  a  rectangle  having  one 
of  its  sides  indefinitely  small,  its  centre  of  gravity  will  be 
situated  in  the  middle  of  the  line  MM'  ;  and  the  ordinate  of 
this  point  will  therefore  be 

PM'  +  iMM'=y'  +  i(y-2<')=i(2/+y'); 
hence,  the  moment  of  this  element  with  reference  to  the  axis 
of  X  will  be 

\{y^y'){y—y')dx=\{y  ^  —y'^)dx  ; 
and  the  moment  with  reference  to  the  axis  of  y  will  be 

^{y—y')d^- 

Thus,  if  Xj  and  y,  denote  the  co-ordinates  of  the  centre  of 
gravity  of  the  entire  surface,  their  values  will  become  known 
from  the  equations 

xx,=fx{y—y')dx, 

>^yi^My'-y")dx. 

186.   To  find  the  centre  of  gravity  of  a  surface  of  revolution. 

Let  the  surface  be  supposed  generated  by  the  revolution  of 
the  curve  AM  {Fig.  85)  about  the  axis  of  x.  The  element  of 
the  surface,  or  the  zone  generated  by  the  elementary  arc  Mwz, 


94  STATICS. 

will  be  expressed  by  2iryds  :  hence,  by  calling  x  the  entire 
surface,  we  shall  obtain 

x-=f2Tryds. 
But  since  the  centre  of  gravity  is  evidently  situated  on  the 
axis  of  revolution,  the  co-ordinate  x,  will  be  alone  necessary. 
To  determine  its  value,  we  take  the  sum  of  the  moments 
with  reference  to  the  plane  yz,  which  sum  being  equal  to  the 
moment  of  the  whole  surface  supposed  concentrated  at  its 
centre  of  gravity,  we  find 

xx=fxy,2fryds] 
whence, 

f2fryxds 
x=       -        ; 

substituting  for  x  and  ds  their  respective  values,  and  suppress- 
ing the  factor  27f  common  to  both  terms  of  the  fraction,  we 
obtain  for  the  absciss  of  the  centre  of  gravity, 
fxy^{dx^  +dy^) 


X.— 


(98). 


~fyV{dx^-^dy^) 
187.  For  the  purpose  of  applying  this  formula,  let  it  be 
required  to  determine  the  centre  of  gravity  of  the  surface  of 
a  spheric  segment.  This  surface  being  generated  by  the 
revolution  of  a  circular  arc  BC  {Fig.  86)  about  the  axis  of 
a:,  we  may  eliminate  one  of  the  variables  in  the  preceding 
formula  by  means  of  the  equation  of  the  circle  ^ 

which  gives,  by  differentiation, 

,  ,     x'^dx'' 

^        y^ 
hence, 

This  value  being  substituted  in  the  integrals  of  equation  (98), 
we  find 

fxy^{dx'^  +  dy'  )  =frxdx  —  \  rx'  +  C, 
fy^{dx'-\-dy^  )=frdx  =rx-\-C'. 
Taking  the    integrals  between  the  limits  a;=AD=a,   and 
x==AB=r,  we  obtain 


CENTRE    OF   GRAVITY.  96 

Sy^idx^  -\-dy^)=r{r—a). 
These  values  transform  the  equation  (98)  into 

x,^\{r-{-a)=a-{-\{r—a)  ; 
thus,  the  centre  of  gravity  is  situated  at  the  middle  of  the 
line  DB. 

188.  To  find  the  centre  of  gravity  of  a  solid  of  revolution 
Mjhounded  hy  tivo planes perpendictdar  to  the  axis,  (Fig:87). 

The  centre  of  gravity  being  necessarily  situated  upon  the 
axis  of  revolution,  which  is  supposed  to  coincide  with  the 
axis  of  X,  it  will  be  sufficient  to  determine  its  absciss  x,. 
The  element  of  the  solid  is  expressed  by  Try  dx,  and  we 
therefore  have 

M=fry*dx (99). 

The  moments  being  taken  with  reference  to  the  plane  ofy,z, 
we  shall  obtain 

Mx=f7ry''xdx (100)  ; 

and  by  dividing  this  equation  by  the  preceding,  we  find 

We  must  eliminate  one  of  the  variables  in  this  formula,  by 
means  of  the  equation  of  the  curve,  and  then  integrate  be- 
tween the  limits  a;=AP  and  a.-=AQ,. 

189.  This  formula  being  applied  to  the  determination  of 
the  centre  of  gravity  of  a  cone,  it  will  be  necessary  to  obtain 
the  two  integrals 

fy'  dx  and  fxy^  dx. 

Eliminating  y^  by  the  equation  of  the  generatrix  y=ax,  we 
obtain,  after  integration, 

fy^  dx—fa^x^  dx=^^, 

-,       -        ,      «^a* 
fy^xdx=fa^x^dx=—j—. 

There  are  no  constants  introduced  by  integration,  since  tlie 
volume  is  equal  to  zero  at  the  origin  A  (Pig:  88).  These 
values,  being  substituted  in  the  formula  (101),  give 


96  STATICS. 

a-x* 

3 
from  which  we  conclude  that  the  centre  of  gravity  of  a  cone 
is  at  a  distance  from  the  vertex  equal  to  three  four  tits  of  the 
altitude  Ax. 

190.  As  a  second  example,  let  the  required  centre  of  gravity 
be  that  of  the  volume  of  a  paraboloid  generated  by  the  revo- 
lution of  the  parabolic  arc  AM  {Pig.  85)  about  the  axis  Ax. 
The  equation  of  the  curve  being  y^  =px,  we  have 

fy'^  dx  =fpxdx  =  \px^ , 
fy- xdx=fpx^ dx=-\px^  : 
these  values  substituted  in  formula  (101),  give 

x,='^^^  =  ^x. 
\px^      ' 

The  constants  introduced  by  integration  are  equal  to  zero  in 
the  present  instance,  for  the  reasons  assigned  in  the  preceding 
paragraph. 

191.  Let  the  solid  of  revolution  be  an  ellipsoid,  the  equa- 
tion of  whose  generatrix  is 

this  value  ofy^  being  substituted  in  the  integrals  of  equation 
(101),  we  obtain,  since  the  constants  are  equal  to  zero, 

/y^dx=—/  (a^dx — x'^dx)  =  —  la^x——-j, 

r,   J       ^"  /?  ,    J         ,j  N     h""  (a'^x^     x^\ 
ly^xdx=—:;  l{a^xdx—x^dx)=-^  I  — ^ -r-  I . 

These  values  reduce  equation  (101)  to 

_\a'X — \x^  _&a'^x — 3x2 
'~  a^   — ix2      12a2— 4x-  ' 
and  by  taking  the  integral  between  a;=0  and  x=a,  we  find, 
for  the  absciss  of  the  centre  of  gravity  of  the  semi-ellipsoid, 

x,  =  ^a. 

192.  To  find  tlie  ceîitre  of  gravity  of  a  volume  generated 


CENTROBARYC   METHOD.  Ô7 

by  the  revolution  of  an  area  embraced  by  a  curve  BMCM' 
{Fig>  89)  about  the  axis  of  x,  this  axis  being  situated  entirely 
without  the  curve. 

Represent  by  y  and  /  the  ordinates  MP  and  M'P  :  the 
volume  generated  by  the  revolution  of  the  element  Mot',  will 
be  equal  to  the  difference  of  the  volumes  generated  by  the 
elementary  rectangles  Mp  and  M'p  ;  thj  expressions  for  these 
volumes  being  ^y'^dx  and  vy'^dx,  that  of  the  element  of  the 
solid  will  be  7r(y2 — y'^)dx]  hence,  if  we  denote  by  M  the 
entire  volume  of  the  solid  generated,  we  shall  have 
M=-!rf{y''  —y"')dx. 

By  taking  the  moments  with  reference  to  the  plane  of  y,  z,  we 
obtain 

Ma:, = vrfiy  '^  —y'^)  xdx. 
The  value  of  x^  will  be  alone  necessary,  si  ice  the  centre  of 
gravity  must  be  situated  on  the  axis  of  abscisses. 

Of  the  Centrobaryc  Method. 

193.  Let  X,  and  y,  represent  the  co-ordinates  of  the  centre 
of  gravity  of  a  plane  surface  MPP'M'  {Fig.  90),  the  area  of 
which  is  represented  by  x.  The  moment  of  the  element  of 
this  surface,  taken  with  reference  to  the  axis  of  re,  is,  by 
Art.  182,  {yxydx  ;  and  by  making  the  sum  of  the  moments 
of  all  the  elements  equal  to  the  moment  of  the  whole  body 
supposed  concentrated  at  its  centre  of  gravity,  we  have 

f\y^dx=y^x. 

The  two  members  of  this  equation  being  multiplied  by  the 
quantity  25r,  it  becomes 

f'>Fy^dx=2-7ryi\: 
The  expression  f^ry^  dx  represents  the  volume  generated  by 
the  revolution  of  the  given  surface  about  the  axis  of  x,  and  the 
second  member  2xy/  is  the  product  of  the  generating  surface 
by  the  circumference  described  by  the  centre  of  gravity; 
hence,  we  deduce  this  general  theorem  :  The  volume  of  every 
solid  of  revolution  is  equal  to  the  product  of  the  generating 
wea  by  the  circuTnference  described  by  its  ceidre  of  gravity. 

194.  Let  it  be  required,  for  example,  to  determine  the 

G  9 


98  STATICS. 

volume  of  the  solid  generated  by  the  revolution  of  an  isosceles 
triangle  ABC  {Fig.  91)  about  the  axis  of  x.  Denote  CD 
by  A,  and  AB  by  a  ;  the  generating  area  will  then  be  ex- 
pressed by  ia/i.  But  the  centre  of  gravity  of  the  generating 
triangle  being  at  a  distance  from  C  equal  to  fCD,  the  circum- 
ference described  by  this  point  will  be  |/iX2îr.  Hence,  the 
volume  will  be  expressed  by  the  product  |/iX2jrX^a/i  = 
fw-a/i-. 

As  a  second  example,  let  us  determine  the  volume  of  a 
right  cone  generated  by  the  revolution  of  the  right-angled 
triangle  ABC  [Fig.  92)  about  the  line  AB.  The  area  of  the 
generatrix  will  be  iAJBxBC.  The  line  CE  being  drawn  to 
the  middle  of  the  side  AB,  the  centre  of  gravity  G  of  the 
generating  area  will  be  situated  upon  this  line  at  a  distance 
from  the  point  E  equal  to  iEC  (Art.  172)  ;  its  ordinate  GD 
will  therefore  be  determined  by  the  proportion 
3  :  1  :  :  EC  :  EG  :  :  CB  :  GD  ; 
whence, 

GD=iCB. 
The  path  described  by  the  centre  of  gravity  will  therefore  be 
expressed  by  f^rxCB;  which,  multiplied  by  the  area  of  the 
generating  triangle  gives  the  volume  of  the  cone  equal  to 
|«-xCB=^  XiAB=:iABx^xCB2. 

195.  Again,  let  the  volume  be  that  of  a  right  cylinder  : 
the  ordinate  GE  of  the  centre  of  gravity  of  the  generating 
rectangle  {Mg.  93)  being  equal  to  ^AC,  the  path  described 
by  this  point  will  be  ;rAC.  This  expression  being  multiplied 
by  the  generating  area  which  is  equal  to  AB  X  AC,  we  have 
srxAC^  xAB  for  the  volume  of  the  cylinder. 

196.  The  area  of  any  surface  of  revolution  may  be  found 
by  a  rule  analogous  to  the  preceding.  For,  if  we  consider 
the  surface  generated  by  the  revolution  of  any  curve  MN 
[F\g.  94)  about  the  axis  of  abscisses,  and  denote  by  y,  the 
ordinate  of  its  centre  of  gravity  G,  we  shall  have,  by  Art. 
178, 

fy^{dx"-  +c?y=*)=y,Xarc  MN (102)  ; 

and  by  multiplying  each  member  by  2v,  this  equation  be- 
comes 


MACHINES — CORDS.  99 

f27ry^{dx^  +  dy')=2vy,  X  arc  MN. 
The  expression  f2Try^{dx^-\-dy'')  representing  the  area  of 
the  surface  generated,  we  conclude,  that  the  area  of  a  surface 
of  revolution  is  equal  to  the  product  of  the  generating  arc 
by  the  circumference  described  by  its  centre  of  gravity. 

197.  Thus,  to  determine  the  surface  of  a  conic  frustrum 

generated  by  the  revolution  of  the  right  line  CD  {Mg.  95) 

about  the  axis  of  x,  we  have  the  ordinate  EG  of  the  centre  of 

,  ,    AC+DB         ,  „       AC+DB  ,  ,    ,, 

gravity  equal  to  ^ ;  and  25rX ^ equal  to  the 

circumference  described  by  this  point  :  hence,  the  product  of 
this  expression   by  the  length  of  the  generatrix  CD  gives 

271-  X —  X  CD=25r .  GE .  CD  for  the  convex  surface  of 

the  conic  frustrum. 

198.  The  two  preceding  theorems  may  be  included  in  a 
single  enunciation,  viz.  :  Every  solid  or  surface  of  revolution 
is  equal  to  the  product  of  its  generatrix  by  the  circumference 
described  by  the  centre  of  gravity  of  the  generatrix. 

Machines. 

199.  Machines  serve  to  transmit  the  action  of  forces  in 
directions  different  from  those  in  which  the  forces  are  applied, 
and  to  modify  the  effects  of  those  forces. 

The  force  applied  to  a  machine  is  called  the  power,  and 
that  which  tends  to  oppose  the  effect  of  the  power  is  called 
the  resistance. 

The  most  simple  machines  are  the  cord,  the  lever,  and  the 
inclined  plane.  To  these  are  sometimes  added  the  pulley, 
the  wheel  and  axle,  the  screw,  and  the  wedge,  which  may 
be  formed  by  very  simple  combinations  of  the  first  three. 
These  machines  are  usually  called  the  Mechariical  Powers. 

Cords. 

200.  We  shall  adopt  the  hypothesis  that  cords  are  perfectly 
flexible,  that  they  are  inextensible,  without  weight,  and  re- 
duced to  their  axes.     If  the  extremities  of  a  cord  be  sohcited 

G2 


100  STATICS. 

by  two  equal  forces  P  and  Q,  {Pig.  96),  Avhich  tend  to  stretch 
it,  the  tension  of  the  cord  will  be  measured  by  one  of  these 
forces  ;  for, since  the  equilibrium  subsists,  we  may  regard  A, 
the  middle  of  the  line  PQ,,  as  a  fixed  point,  and  drop  the  con- 
sideration of  that  portion  of  the  cord  included  between  A  and 
Q,  ;  thus,  the  force  P,  acting  alone  against  the  fixed  point  A, 
will  measure  the  tension  of  the  cord  PQ,. 

201.  When  the  force  Q,  exceeds  P,  a  portion  of  Q.  equal  to 
P  is  employed  to  stretch  the  cord,  while  the  remaining  part 
of  the  force  tends  only  to  move  the  cord  in  the  direction  from 
P  towards  Q,  :  thus  the  tension  will  be  measured  by  the  least 
of  these  forces. 

202.  If  three  cords  be  united  by  a  knot,  the  conditions  of 
equilibrium  are  similar  to  those  which  obtain  when  any  three 
forces  act  on  a  point.  The  force  acting  in  the  direction  of 
each  cord  must  be  equal  and  directly  opposed  to  the  resultant 
of  the  other  two  ;  hence,  the  conditions  of  equilibrium  require 
that  the  three  forces  be  situated  in  the  same  plane,  and  bear 
to  each  other  the  following  relations  {Fig>  97), 

P  :  Q,  :  R  :  :  sin  7>  :  sin  g-  :  sin  r. 

203.  This  proportion  will  be  insufficient  to  establish  the 
equilibrium,  if  the  cords  are  united  by  a  sliding  knot.  For, 
by  regarding  P  and  R  as  fixed  points  {Fig.  98),  to  which  the 
cord  PCR  is  attached,  if  the  force  Q,  be  supposed  to  act  upon 
this  cord  by  means  of  a  ring  or  sliding  knot,  the  point  C  will 
describe  an  ellipse,  the  plane  of  which  will  pass  through  the 
points  P  and  R.  But  the  revolution  of  this  ellipse  around 
the  axis  PR  will  generate  an  ellipsoid,  having  its  transverse 
axis  equal  to  PC  |-CR,  and  the  point  C  will  necessarily  be 
found  upon  the  surface  of  the  ellipsoid,  or,  in  other  words, 
at  some  point  of  the  moveable  ellipse  ;  but  the  point  C  being 
only  subject  to  motion  when  the  force  Q.  has  a  component  in 
the  direction  of  the  elliptical  arc,  the  equilibrium  will  be 
maintained  when  the  direction  of  the  force  Q,  is  normal  to 
the  ellipse.  If  the  line  T^  be  drawn  tangent  to  the  curve, 
wo  shall  have,  from  the  well  known  property  of  the  ellipse, 

Z.TCP=ZRC^; 
and  by  subtracting  these  angles  from  the  right  angles  TON, 
if  ON,  there  will  remain 


CORDS.  •  101 

z:pcn=zncr; 

thus  the  angle  PCR  must  be  bisected  by  the  direction  of  the 
force  Ct,  and  the  proportion 

P  :  R  :  :  sin  NCR  :  sin  PCN 
becomes,  in  the  present  case, 

P  :  R  :  :  sin  NCR  :  sin  NCR; 
whence,  P  and  Q,  are  equal  to  each  other. 

204.  The  funicular  machine  consists  of  a  number  of  cords 
united  to  each  other  at  several  knots,  and  maintaining  an 
equilibrium  between  the  forces  applied  to  these  cords. 

205.  When  several  forces  P,  R,  S,  T,  &c.  (Fig:  99),  act 
conjointly  at  a  single  knot,  their  number  will  be  reduced 
by  unity,  if  we  substitute  for  any  two  forces  P  and  R  their 
resultant  R';  and  by  a  repetition  of  the  same  process  the 
entire  system  may  always  be  reduced  to  three  forces  united 
at  a  single  knot. 

206.  Let  there  be  several  forces  P,  P',  P",  F",  P'^,  (fee. 
{Fig.  100),  acting  at  the  knots  A,  B,  C,  &c.  of  the  cord  ABC. 
The  conditions  of  equilibrium  of  these  forces  may  be  reduced 
to  those  of  a  system  acting  on  a  single  point  :  for,  let  R 
represent  the  resultant  of  the  forces  P  and  P'  ;  since  its  effect 
must  be  destroyed  by  the  third  force  acting  in  the  line  AB, 
the  direction  of  this  resultant  must  coincide  with  the  pro- 
longation of  AB  :  but  the  point  of  application  of  a  force  may 
be  assumed  any  where  on  its  line  of  direction,  and  hence  we 
may  transfer  the  force  R  to  the  point  B.  If  it  be  there  de- 
composed into  two  components  parallel  and  equal  to  P  and 
P',  the  effect  will  be  the  same  as  if  the  two  forces  P  and  P' 
had  been  transported  parallel  to  their  original  directions,  and 
applied  at  the  point  B.  In  hke  manner,  by  transporting  the 
forces  P,  P',  P",  (fcc,  which  are  supposed  to  be  applied  at  B, 
to  the  point  C,  the  entire  system  may  be  considered  as  acting 
on  this  point.  Thus  the  conditions  of  equilibrium  are, 
(Art.  54), 

2(P  cos  «)  =0,    2(P  cos  /3)  =0,    s(P  cos  y)=0. 
To  determine  the  ratio  of  the  extreme  tensions  P  and  P,^, 
we  will  denote  by  t  and  t'  the  tensions  of  the  portions  AB  and 
BC,  and  by 


102  •  STATICS. 

a  the  angle  PAP',     a'  the  angle  ABP",     a"  the  angle  BCP'", 
b  the  angle  P'AB,     b'  the  angle  P"BC,     b"  the  angle  P"'CP"  ; 
we  shall  then  obtain,  Art.  202, 

P  :  ^  :  :  sin  6  :  sin  a, 

t  :  i'  :  :  sin  b'  :  sin  a', 

i  :  P"  :  :  sin  b"  :  sin  a"  ; 

whence,  by  multiplication,  suppressing  the  factors  which  are 
common  to  the  two  first  terms,  we  have 

P  :  P"  :  :  sin  6  xsin  6'xsin  b"  :  sin  a  Xsin  a'  Xsin  a". 
We  may,  in  like  manner,  determine  the  relations  between 
aiiy  other  two  forces. 

207.  If  the  forces  P',  P",  P'",  (fcc.  be  supposed  parallel,  we 
shall  have 

b+a'=lSO%     6' +  «"=180°; 

and  since  the  sine  of  an  angle  is  eaual  to  the  sine  of  its  sup- 
plement, we  must  have 

sin  6= sin  «',     sin  b'—sin  a"  ; 
and  the  preceding  proportion  will  then  reduce  to 

P  :  P"  :  :  sin  b"  :  sin  a. 
If  the  forces  P',  P",  and  F"  represent  weights  {Mg.  101),  the 
entire  system  will  be  situated  in  the  same  vertical  plane  ;  for, 
the  right  line  AF  being  vertical,  the  plane  of  the  forces 
P,  P',  and  t  will  be  vertical.  For  a  similar  reason,  the  plane 
of  the  forces  t,  P",  and  t',  will  be  vertical  ;  but  the  line  AB 
not  being  vertical,  it  is  impossible  to  pass  more  than  one  ver- 
tical plane  through  it  :  hence,  the  forces  P,  P',  t,  P",  and  i' 
will  be  situated  in  the  same  vertical  plane.  The  same  rea- 
soning may  be  extended  to  a  greater  number  of  forces. 

208.  The  extreme  forces  P  and  F"  being  required  to  sus- 
tain the  resultant  of  all  the  others,  this  resultant  must  be 
directly  opposed  to  that  of  the  forces  P  and  P",  and  must 
consequently  f>ass  through  the  point  G,  at  which  the  direc- 
tions of  those  forces  intersect.  Moreover,  its  direction  must 
be  vertical,  being  parallel  to  the  components  P',  P",  and  P"', 
and  it  will  therefore  be  represented  by  the  vertical  line  GH 
drawn  through  the  point  G. 

209.  If  we  regard  a  heavy  cord  as  a  funicular  polygon, 


CATENARY.  •  103 

loaded  with  an  infinite  number  of  small  weights,  it  results 
from  what  precedes  that  the  effect  produced  on  the  fixed  points 
by  the  weight  of  the  cord  may  be  estimated  by  drawing  the 
tangents  PG  and  QG  {Fig.  102),  and  applying  at  G  a  weight 
equal  to  that  of  the  cord  ;  since  if  we  denote  this  weight  by 
G,  we  shall  then  have 

P  :  Gl  :  G  :  :  sin  LGa  :  sin  LGP  :  sin  PGQ. 

Of  the  Catenary. 

210.  The  catenary  is  the  curve  which  a  perfectly  flexible 
cord  assumes  when  it  is  suspended  from  two  fixed  points 
A  and  B  {Fig.  103),  and  subjected  to  the  action  of  the  force 
of  gravity.  We  will  suppose  that  the  cord  is  uniformly 
heavy,  and  that  the  force  of  gravity  is  exerted  on  every 
particle  :  it  will  readily  appear,  as  in  Art.  207,  that  the  curve 
will  be  situated  in  a  vertical  plane.  Let  the  origin  of  co- 
ordinates be  assumed  at  A,  the  horizontal  line  AC  being  the 
axis  of  abscisses  ;  the  co-ordinates  of  a  point  M  will  then  be 
AP=a:,  and  PM=y.  Through  the  point  M,  and  through  the 
origin  A,  let  tangents  AH  and  MH  be  respectively  drawn, 
intersecting  at  the  point  H,  and  through  this  point  draw  the 
vertical  line  HL.  If  we  consider  the  portion  of  the  cord 
MA,  we  shall  have,  by  Art.  209, 

tension  at  A  :  weight  of  the  portion  AM  :  :  sin  LHM  :  sin  AHM (103). 

Let  s  denote  the  length  of  the  arc  AM  ;  A  the  tension  of  the 
cord  at  the  point  A,  which  is  exerted  in  the  direction  of  the 
tangent  AH  ;  and  «  the  angle  included  between  this  tangent 
and  the  horizontal  line  AC.  The  quantities  A  and  «  will 
remain  constant. 

The  tension  at  A,  being  a  quantity  of  the  same  kind  as  that 
contained  in  the  second  term  of  the  preceding  proportion, 
will  necessarily  be  expressed  by  a  weight  ;  and  if  we  repre- 
sent by  p  the  weight  of  a  portion  of  the  cord  whose  length  is 
equal  to  unity,  sp  will  express  the  weight  of  the  part  AM, 
and  the  tension  at  A  will  be  of  the  form  ap.  Thus  the  two 
first  terms  in  the  above  proportion  will  be  replaced  by  the 
ratio  ap  :  sp,  or  by  its  equal  a  :  s]  hence, 

a  :  5  :  :  sin  LHM  :  sin  AHM (104). 


104  •  STATICS. 

211.  To  determine  the  analytical  expressions  for  the  sines 
which  enter  into  this  proportion,  we  remark,  that  in  the 
elementary  triangle  7nMn,  we  have 

Mm  X  sin  mMn =mn,     Mm  X  cos  m'M.7i=M.n  ; 
or, 

__        mji  _^        M;t 

sm  mMfi=  ^r-r— 5     cos  mM?i=r  — —  ; 

Mm'  Mm  ' 

and  replacing  these  elementary  lines  by  their  analytical  values, 
these  equations  become 

dx  di/ 

sin  ?n'M.fi——r-,     cos  nïM.n=—r (105). 

as  as 

Bat  the  angle  mMn  included  between  the  vertical  and  the  arc 
of  the  curve,  is  equal  to  the  angle  LHK  formed  by  the 
vertical  with  the  tangent  at  M  ;  hence, 

sin  LHK=4^ ,     cos  LHK=4^ (106). 

as  as 

The  first  of  these  equations  may  be  reduced  to 

sinLHM— (107); 

as 

for  the  angles  LHK  and  LHM  being  supplements  of  each 
other,  we  have 

sin  LHK=sin  LHM. 
Again,  the  angles  AHK  and  AHM  being  supplements  of  each 
other,  we  obtain 

sin  AHM^sin  AHK=sin  (LHK-LHA)  ; 
and  from  the  well  known  trigonometrical  formula  for  the  sine 
of  the  difference  of  two  angles,  we  have 

sin  AHM=sin  LHK  cos  LHA— sin  LHA  cos  LHK  ; 
eliminating  sin  LHK  and  cos  LHK  by  means  of  the  equations 
(106),  we  find 

sin  AHM^'^-^cos  LHA-^'sin  LHA (108). 

as  as 

The  triangle  LAH  being  right-angled  at  L,  the  angles  LHA 
and  HAL  are  complements  of  each  other,  and  the  latter  hav- 
ing been  denoted  by  a,  we  obtain 

cos  LHA=sin  «,    sin  LHA=cos  «. 


CATENARY.  105 

212.  These  values  substituted  in  equation  (108)  give 

»        sin  AHM=^sin  «-"^cos  « (109)  ; 

as  ds 

and  the  equations  (107)  and  (109)  convert  the  proportion  (104) 

into 

dx    dx  .  dy 

a:  s  '.:  -y  :  -y-sin  « — fcos  *, 
as     ds  ds 

From  this  proportion  we  deduce  the  equation 

5=asin« — a-^cos» (110). 

dx 

This  equation  contains  three  variables,  one  of  which  may 
be  eliminated  by  means  of  the  relation 
ds=^y/{dx'i-\-dy^). 
For,  by  differentiating  equation  (110),  regarding  dx  as  con- 
stant, we  find 

ds= — a  cosa— -^  ; 
dx 

and  by  equating  these  values  of  ds,  and  dividing  each  mem- 
ber of  the  equation  by  dx,  we  obtain 

\/(l  +  -r^  )=— a  cos«— ^, 
^    \      dx^/  dx""' 

or,  by  division, 

d'^y 

— a  cos«~ 

dx^ 


v/O+g) 

This  equation  will  become  integrable,  if  we  multiply  its  two 
members  by  2dy  ;  we  shall  thus  obtain 

2dy^^ 
2dy— — a  cos  «__ ^  ;  ^  4.  L 

whence,  by  integration,  jV^'^ 

y=-acos«^(l-F^)+c. 

This  equation  being  multiplied  by  dx  gives 

{c—y)dx=a  cos*y/{dx^-\-dy'^)  ; 


106  STATICS. 

and  by  reduction 

dy^^/[{c-yy—a^  cos»c«] .^^^^ 

dx  a  cos  « 

213.  The  constant  c  may  be  determined  by  the  consider- 
ation that  at  the  point  A, 

x=0,    y=0,    and  ^=tang«. 
dx 

These  values  reduce  the  equation  (111)  to 


tang«  — ^  ^                      i.; 
a  cos« 

from  which 

we  deduce 

but 

a  tangae  cos  »=^{c^—a^  cos*  «) : 

whence, 

tang*  cos  «= sin  a; 

a^  sin^  *=c2  —  a^  cos^  *, 

and  consequently 

c^=a2  (sin'  a+cos^  «)=a'. 
Thus  the  constant  c  is  equal  to  a,  and  by  substituting  its 
value  in  equation  (111),  we  find  for  the  differential  equation 
of  the  catenary, 

dy_^[{a—yY-a^  cos^  a\ ,^^^y 

dx  a  cos  <* 

214.  It  appears  from  a  comparison  of  this  equation  with 
(110),  that  the  catenary  curve  is  rectifiable  ;  for,  if  the  pre- 
ceding value  of  Y  ^^  substituted  in  equation  (110),  we  shall 

obtain 

s=a  sin  a— ^[{a—yY  —a^  cos^»] (113)  : 

from  this  expression  the  value  of  s  may  be  readily  found  in 
terms  of  y,  when  the  constants  a  and  »  have  been  determined. 
.  215.  To  integrate  the  differential  equation  of  the  catenary, 
we  make 

a—y=z,     a  cos  et=b (114)  ; 

and  we  then  obtain 

dy  =  — dz  ] 
these  values  substituted  in  equation  (112)  give 


MACHINES. 


107 


dx  =  --4^—— (115); 

this  expression  becomes  integrable  by  making 

^{z^-h^)=z—t (116): 

which  by  squaring  and  reducing,  gives 

By  the  differentiation  of  this  equation,  we  obtain 

xdt-\-tdz=tdtj 
or, 

dz  _      dt 

z—t~~T' 

This  relation,  in  connexion  with  that  assumed  above  (116), 
converts  the  equation  (115)  into 

,      bdt 
dx  —  — J 
t 

which  gives,  by  integration, 

x=-h  log^+e; 
and  by  substituting  for  t  its  value  expressed  in  terms  of  sr,  we 
obtain 

x=h\o%\z—^{z''—h^-)\^e\ 

or  finally,  by  replacing  the  quantities  h  and  z,  by  their  values 

given  in  equations  (114),  we  find 

ar=acos<«log|a— y— ^[(a— y)  =  — a^  cos2<«]|  +e (H^). 

216.  To  determine  the  value  of  the  constant  e,  we  observe 
that  at  the  point  A,  a;=0,  and  y=0  ;  which  conditions  reduce 
the  equation  (117)  to 

e=— «coscelog  |a[l— y/(l— C0S2  «)]|. 
This  value  substituted  in  equation  (117)  gives 

a;=acos«log[a— y— -v/(a— y)"— a^cos*  «  ] 
—a  cosee  log  [a(l  — v/l— cos=  «)]  ; 
or  by  reduction. 

Such  is  the  equation  of  the  catenary. 

217.  The  values  of  the  constants  c  and  e  have  been  deter- 
mined in  functions  of  a  and  «f  ;  but  these  two  quantities  are 


108 


STATICS. 


Still  unknown.  To  determine  their  values,  we  will  suppose 
that  jc'  and  y'  represent  the  known  co-ordinates  of  the  second 
point  of  suspension  B,  and  I  the  length  of  the  curve  AMB  ; 
these  values  being  substituted  in  the  equations  (113)  and 
(118),  we  obtain 

l=a  sin  »— ^[{a—y'Y  —a^  cos*  «], 


X  —a  cos«» 


^\  â[l— /(l-cos^«)]  ) 


218.  These  equations,  in  connexion  with  the  relation 
C0S2  «+sin'  «=1, 
determine  the  values  of  a,  cos  «,  and  sin  ec,in  functions  of^.y', 
and  /.  But  another  difficulty  still  presents  itself;  this  consists 
in  the  proper  choice  of  the  signs  with  which  to  affect  cos  «, 
and  the  radicals  which  in  the  preceding  expressions  have  not 
received  the  double  sign.  To  resolve  this  difficulty,  we  will 
determine  the  co-ordinates  of  that  point  to  which  the  max- 
imum ordinate  appertains.     The  characteristic  property  of 

this  point  is  that  -^=0,  which  reduces  equation  (112)  to 
dx 

^/[{a-yy—a''  cos''  ^]^Q . 
acos«  ' 

and  consequently, 

a  — 2/=acos« (H^)- 

To  establish  the  condition  that  this  equation  belongs  to  a 
maximum,  rather  than  to  a  minimum  value,  we  attribute  the 

proper  sign  to  the  second  differential  co-efficient  -j-^. 

But  by  squaring  the  equation  (112),  we  obtain 
dy'^  _{a—yY  —a^  cos»  « 
dx'^  a^  cos 2  a.  ' 

and  by  differentiating,  and  dividing  each  member  by  2c?y,  we 
find 

d^y__  a— y     . 
dx^         a'cos'^tt'^ 
substituting  in  this  equation  the  value  oia—y  determined  in 
equation  (119),  we  obtain 

dry^^ l__ 

dx'  a  cos  <*' 


LEVER.  109 

219.  This  equation  indicates  that  the  condition  of  a  maxi- 
mum will  be  fulfilled  by  attributing  the  same  sign  to  a  and 
cos  «  ;  but  these  signs  must  be  positive  ;  for,  if  they  were 
negative,  the  value  of  y  determined  hj  the  equation  (119) 
would  be  also  negative,  which  is  evidently  inadmissible  in 
the  hypothesis  adopted,  that  the  positive  ordinates  are  reckoned 
from  the  line  AC  downwards.  From  the  equation  (119)  we 
likewise  infer,  that  the  quantity  a  exceeds  the  maximum 
value  of  y,  and  therefore  that  it  exceeds  all  other  values. 
Let  EF  represent  the  maximum  ordinate  {Fi^.  103);  it  is 
evident  that  between  the  limits  â;=0  and  a-=AE,  as  y  in- 
creases, the  arc  of  the  catenary  will  likewise  increase.  But 
it  appears  from  equation  (113)  that  the  increase  of  y  will  not 
necessarily  involve  that  of  the  arc  5,  unless  the  radical  in  that 
formula  be  affected  with  the  negative  sign.  For,  as  y  in- 
creases, the  quantity  a — y  will  decrease,  and  the  value  of  the 
radical  will  therefore  decrease  ;  but  the  smaller  the  value  of 
this  radical,  the  less  it  will  diminish  the  positive  part  of  the 
expression  a  sin  «,  and  the  greater  will.be  the  value  of  the 
arc.  The  equation  (113)  is  therefore  in  perfect  accordance 
with  the  hypothesis  that  the  co-orJinate  has  not  attained  its 
maximum  value.  But  from  :2;=AE  to  a;=AD,  the  arc  5 
should  increase  while  y  diminishes,  and  since  this  decrease 
in  the  value  of  y  augments  the  value  of  the  radical  expres- 
sion, the  required  condition  can  only  be  fulfilled  by  affecting 
the  radical  with  the  positive  sign  :  thus,  between  the  limits 
ar=AE  and  ar=AD,  the  sign  of  the  radical  must  be  changed 
in  the  formula  (113). 

Of  the  Lever. 

220.  The  lever  is  a  bar  of  wood  or  metal  moveable  around 
a  fixed  point,  which  is  called  the  fulcrum.  To  simplify  the 
considerations  which  relate  to  this  machine,  we  shall  regard 
the  lever  as  destitute  of  thickness,  and  will  therefore  represent 
it  by  a  simple  line,  either  straight  or  curved.  Let  a  lever  AB 
{Fig.  104)  be  sohcited  by  the  two  forces  P  and  P'  ;  the  effect 
of  these  forces  cannot  be  destroyed  by  the  resistance  of  a  fixed 

10 


110  STATICS. 

point  C,  unless  they  are  situated  in  a  plane  passing  through 
this  point.  If  this  condition  be  fulfilled,  the  equilibrium  will 
be  maintained,  when  the  sum  of  the  moments  taken  with 
reference  to  the  point  C  is  equal  to  zero. 

221.  If  the  lever  is  capable  of  sliding  along  its  point  of 
support,  it  will  also  be  necessary  that  the  resultant  of  the 
forces  acting  on  the  lever  should  be  perpendicular  to  the  lever 
at  the  point  of  support. 

222.  When  the  lever  is  straight  and  the  two  forces  parallel 
to  each  other,  ifp  and  p'  represent  the  lengths  of  the  portions 
AC  and  BC  {Pig.  106),  we  shall  have  from  the  theory  of 
parallel  forces  (Art.  73), 

V  :V'  ::p'  :p] 
from  which  we  infer,  that  when  the  forces  are  in  equilibrio, 
their  intensities  will  be  inversely  proportional  to  the  arms  of 
the  lever. 

223.  If  the  lever  be  curved,  and  a  right  line  ED  {Fig.  105) 
be  drawn  through  the  fulcrum  C,  the  forces  may  be  conceived 
to  be  applied  at  the  points  E  and  D  taken  on  their  respective 
directions  ;  we  shall  thus  obtain 

P  :  F  :  :  CD  :  CE. 

224.  Levers  are  divided  into  three  kinds.  In  the  first  kind, 
the  fulcrum  C  {Fig.  106)  is  situated  between  the  power  and 
the  resistance  :  in  the  second  kind,  the  resistance  R  {Fig. 
107)  is  situated  between  the  power  and  the  fulcrum  ;  and  in 
the  third  kind  {Fig.  108),  the  power  is  between  the  fulcrum 
and  the  resistance. 

The  balance  and  steelyard  are  examples  of  the  first  kind 
of  lever  ;  a  bar  of  iron  used  in  raising  weights  and  having 
its  fulcrum  at  one  extremity,  forms  a  lever  of  the  second 
kind  ;  the  treddle  of  a  turning  lathe  is  a  lever  of  the  third 
kind. 

225.  The  effect  produced  by  the  weight  of  a  lever  may  be 
readily  estimated  by  regarding  it  as  a  force  S  applied  at  the 
centre  of  gravity  of  the  lever.  For  example,  let  P  and  P' 
{Fig.  109)  be  two  weights  suspended  from  the  extremities  of 
the  lever  AB,  whose  centre  of  gravity  is  situated  at  G  ;  we 


LEVER.  Ill 

shall  have,  by  virtue  of  the  principle  of  the  moments, 

P'xCB+SxCG=PxAa 
This  equation  will  determine  either  P  or  F;  and  the  weight 
sustained  by  the  fixed  point  will  be 

P  +  P'  +  S. 
If  the  power  and  resistance  act  in  opposite  directions,  regard 
must  be  had  to  the  directions  in  which  they  tend  to  turn  the 
lever;   thus,  in  Fig.  110,  the  equation  of  the  moments  be- 
comes 

PXCA  +  SXCG=P'XCB (120); 

and  the  weight  sustained  by  the  fulcrum  is 

P+S-F. 
226.  Let  the  lever  CB  {Fig.  110)  be  supposed  homogeneous, 
and  of  uniform  weight  throughout  its  length  :  represent  by 
tn  the  weight  of  a  portion  of  the  lever  whose  length  is  one 
foot.  If  X  represent  the  length  of  the  lever  expressed  in  feet, 
its  weight  S  will  be  expressed  by  mx^  and  should  be  regarded 
as  a  force  acting  at  its  centre  of  gravity,  which  corresponds 
to  the  middle  point  G  :  thus,  if  we  make  CA=a,  the  equation 
(120)  will  then  become 

Va  +  \xXmx=.V'y.x] 
from  which  we  deduce 

F==— -fima: (121). 

X 

\i,  therefore,  x  be  assumed  arbitrarily,  this  formula  will  make 
known  the  value  of  P'  ;  but  it  may  be  required  to  assign  the 
value  of  X  which  shall  render  P'  the  least  possible  ;  we  must 
then  regard  P'  as  a  function  of  x,  and  make  the  differential     0^ 

co-efficient  -r-  equal  to  zero  ;  we  shall  thus  obtain     ,J^  \ 


dx 
whence, 


—  — 4-im=0;         ^^ 


2Pa  ,  //2Pa 


x'=- 


and  X 


m 


-^m 


By  substituting  this  value  in  equation  (121),  we  obtain 


112 


STATICS. 


P  = 


Va 


or,  by  reduction, 


2Pa 


^     \  in  / 

227.  The  cornmon  balance  is  an  important  application  of 
the  lever.  It  consists  essentially  of  a  lever  having  equal  arms, 
from  the  extremities  of  which  are  suspended  scales  of  equal 
weight.  The  lever  of  the  balance,  which  is  called  the  beam, 
is  sustained  by  a  horizontal  axis  perpendicular  to  its  length, 
which  rests  upon  a  firm  support,  and  the  substance  to  be 
weighed,  being  introduced  into  ne  of  the  scales,  is  counter- 
poised by  the  addition  of  known  weights  in  the  opposite 
scale.  The  figure  of  the  beam  is  so  chosen  that  its  centre 
of  gravity  will  be  found  immediately  beneath  the  axis,  or 
centre  of  motion,  when  the  beam  has  assumed  a  horizontal 
position  ;  and  the  weights  suspended  from  its  two  extremities 
are  known  to  be  equal  when  they  will  retain  the  beam  in 
this  situation.  If  the  centre  of  gravity  were  found  upon  the 
axis,  the  beam  would  obviously  rest  in  any  position,  and  there 
would  be  nothing  to  indicate  the  equality  of  the  weights  io 
the  two  scales  ;  and  if  this  centre  were  situated  above  the 
axis,  the  beam  would  have  a  tendency  to  overturn  if  deranged 
in  the  slightest  degree  from  the  horizontal  position. 

228.  When  the  balance  has  been  constructed  with  such 
accuracy  that  the  lengths  of  the  arms  are  exactly  equal,  the 
beam  will  assume  the  horizontal  position  if  equal  weights  be 
introduced  into  the  two  scales  ;  but  in  the  false  balance,  where 
the  lengths  of  the  arms  are  unequal,  the  weights  necessary  to 
maintain  the  beam  in  this  position  are  likewise  unequal.  In 
this  case,  the  weight  of  the  body  may  be  obtained  by  counter- 
poising it  successively  in  the  tv/o  scales  :  the  true  iceight  will 
be  a  geometrical  mean  between  the  two  ajiparent  weights. 
For  let  2^  and  p  represent  the  lengths  of  the  two  arms,  and 
W  the  true  weight  of  the  body.  Then,  if  a  weight  P,  sus- 
pended from  the  extremity  of  fhe  arm  y>,  be  supposed  to 
sustain  the  weight  W  when  suspended  from  the  extremity 


LEVER.  113 

of  the  arm  jo',  the  conditions  of  equilibrium  in  the  lever 
(Art.  220)  will  give 

But  if  the  weight  W  be  transferred  to  the  extremity  of  the 
arm  j»,  it  will  be  necessary  to  apply  a  dijEFerent  weig-ht  P'  to 
the  extremity  of  the  arm  p',  in  order  that  the  equilibrium 
may  be  preserved.     Thus  we  shall  have 

and  by  multiplying  the  corresponding  members  of  these  two 
equations,  we  obtain 

or,  by  reduction, 

W=y(PP'); 
hence,  the  truth  of  the  proposition  enunciated  becomes  appa- 
rent. 

229.  It  is  frequently  necessary  that  the  balance  employed 
should  possess  great  sensibility,  or  should  be  capable  of  indi- 
cating very  minute  differences  in  the  weights  of  the  substances 
placed  in  the  two  scales.  The  sensibility  of  the  balance  is 
measured  by  the  smallness  of  the  weight  necessary  to  produce 
a  given  inclination  of  the  beam,  when  the  scales  are  charged 
with  a  given  load. 

The  sensibility  depends  upon  the  following  particulars. 

1°.  The  beam  should  be  as  light  as  is  consistent  with  a 
proper  degree  of  strength,  in  order  that  the  friction  at  the 
axis,  which  is  proportional  to  the  pressure,  may  oppose  the 
least  possible  resistance  to  the  motion  of  the  beam. 

For  the  same  reason  the  axis  is  constructed  of  hardened 
steel,  and  has  the  form  of  a  knife-edge,  or  triangular  prism, 
the  lower  edge  of  which  rests  upon  polished  steel  or  agate 
planes. 

2°.  The  lengths  of  the  arms  should  be  as  great  as  possible, 

other  things  remaining  the  same,  since  the  moments  of  the 

weights  introduced  into  the  scales,  taken  with  reference  to 

the  centre  of  motion,  will  be  directly  proportional  to  these 

lengths.     Thus,  the  same  weight,  placed  at  twice  the  distance 

from  the  centre  of  motion,  will  exert  a  double  effort  to  turn 

the  beam. 

H 


114 


STATICS. 


3".  The  sensibility  will  be  increased  by  diminishing  the 
distance  between  the  centre  of  gravity  of  the  beam  and  the 
centre  of  motion.  For,  when  the  beam  has  been  deranged 
from  the  horizontal  position  through  a  given  angle  {Fig:  111), 
the  weight  of  the  beam  W,  which  acts  at  its  centre  of  gravity 
G,  will  exert  an  effort  to  restore  it  to  its  former  position, 
which  effort  will  be  directly  proportional  to  the  moment  of 
the  weight  W,  taken  v  ith  reference  to  the  centre  of  motion 
D;  this  moment  will  be  expressed  by  Wxdg.  But  the 
derangement  of  the  beam  having  been  made  through  a  given 
angle,  the  distance  dg  will  evidently  be  proportional  to  DG, 
the  distance  between  the  centre  of  gravity  of  the  beam  and 
the  centre  of  motion.  Thus,  in  proportion  as  the  distance 
DG  is  diminished,  the  tendency  of  the  weight  of  the  beam  to 
counteract  the  derangement  which  would  be  produced  by  ah 
inequality  of  the  weights  in  the  two  scales  will  likewise  be 
diminished,  or  the  sensibility  will  be  increased. 

4°.  The  line  joining  the  points  of  suspension  of  the  two 
scales  should  pass  through  the  centre  of  motion.  For,  if  the 
centre  of  motion  be  found  at  C  above  the  line  AB,  and  the 
beam  be  supposed  to  have  assumed  the  inclined  position 
represented  in  Fig.  Ill,  the  effective  arm  of  lever  CE'  of  the 
scale  P'  will  evidently  be  greater  than  the  arm  CE  of  the 
scale  P.  Thus  the  beam  may  have  a  tendency  to  return  to 
the  horizontal  position,  although  the  weight  P'  be  less  than  P. 
And  if,  on  the  contrary,  the  centre  of  motion  be  placed  at  a 
point  C  below  the  line  AB,  the  lever-arm  C'F  of  the  scale  P 
will  exceed  that  of  the  scale  P',  and  the  beam  would  therefore 
have  a  tendency  to  overturn,  although  the  weights  in  the 
scales  were  equal  to  each  other.  When  the  centre  of  motion 
is  situated  at  the  point  D,  the  equality  of  the  two  arms  will 
be  preserved,  whether  the  beam  be  in  a  horizontal  or  inclined 
position. 

5°.  The  sensibility  of  the  balance  will  be  increased  by 
diminishing  the  load  with  which  the  scales  are  charged,  since 
the  friction  at  the  axis  will  be  diminished  in  the  same  pro- 
portion. 

230.  A  very  accurate  balance  will  be  sensibly  affected  by 
the  addition  of  jg^  part  of  the  load  with  which  the  scales 
are  charged. 


LEVER.  115 

231.  The  steelyard,  represented  in  Pig.  112,  is  a  balance 
having  unequal  arms,  and  is  so  constructed  that  a  moveable 
weight  P,  applied  successively  at  different  points  of  the  longer 
arm,  shall  sustain  in  equilibrio  different  weights  suspended 
from  the  extremity  of  the  shorter  arm.  The  longer  arm 
GB  is  so  graduated  as  to  indicate  the  weight  which  will  be 
supported  by  the  moveable  weight  P^  when  placed  at  each 
of  these  divisions. 

232.  To  discover  the  law  according  to  which  this  arm 
should  be  graduated,  we  will  denote  by 

W,  W,  W",  &c.,  the  weights  suspended  successively 

from  the  extremity  of  the  shorter  arm  , 
j9,  p',  p",  &c.,  the  corresponding  distances  at  which  the 
weight  P  must  be  placed  to  maintain  the  equilibrium, 
r,  the  length  of  the  shorter  arm  , 
IV,  the  weight  of  the  beam, 

r',  the  distance  of  its  centre  of  gravity  from  the  fulcrum^ 
Then,  if  the  centre  of  gravity  of  the  beam  be  supposed  to  lie 
on  the  side  of  the  longer  arm,  as  usually  happens,  the  con- 
ditions of  equilibrium  will  give 

Wr=wr'  +  P/?, 

Wr—tvr'  +  Fp\ 

W"r=ur'  +  Fp", 

&c.     &c.     &c.  ; 

and  by  subtracting  each  of  these  equations  from  that  which 

follows,  we  obtain 

(W'-W)r=(p'-p)P, 

(W"-W')/-=(p"— _p')P, 
(  W" — W")r = (p'" —//')?• 
If  the  weights  W,  W,  W",  &c.  be  supposed  to  increase  in 
arithmetical  progression,  we  shall  have 

W— W=W"— W'=W"'— W"=&c.  ; 
and  therefore 

2)'  —2i=2}" — p'=p"' — ;y'  =  (fcc.  ; 
thus  the  distances  ^?,  />',  p",  <fcc.  will  likewise  increase  in 
arithmetical  progression. 

If,  for  example,  the  moveable  weight  P  when  placed  at  a 
point  F  should  be  found  to  support  a  weight  of  10  pounds, 

H2 


116  STATICS. 

and  if  when  placed  at  the  point  E,the  weight  supported  should 
be  found  equal  to  20  pounds,  we  might  divide  the  distance  EF 
into  ten  equal  parts,  and  the  points  of  division  will  cor- 
respond to  the  weights  11  pounds,  12  pounds,  13  pounds,  &c. 
The  zero  of  the  scale  will  evidently  be  found  at  that  point 
from  which  the  weight  P  is  suspended  when  it  merely  serves 
to  counterpoise  the  weight  of  the  lever.  The  steelyard  is 
frequently  constructed  in  such  a  manner  that  the  two  arms 
of  the  lever  counterpoise  each  other  :  the  zero  of  the  scale 
will  then  coincide  with  the  fulcrum. 

Of  the  Pulley. 

233.  The  pulley  is  a  wheel  having  a  groove  cut  in  its  cir- 
cumference for  the  purpose  of  receiving  a  cord  which  par- 
tially envelopes  it  :  when  a  motion  is  imparted  to  this  cord  it 
is  immediately  communicated  to  the  pulley,  causing  it  to  turn 
about  an  axis  which  passes  through  its  centre,  and  is  usually 
supported  by  a  curved  piece  of  iron  terminating  in  a  hook 
{Mg.  113). 

Pulleys  are  distinguished  into  two  kinds,  the  fixed  and  the 
moveable.  In  the  fixed  pulley,  the  hook  is  attached  to  an 
immoveable  point,  as  in  {Fig.  113);  and  in  the  moveable 
pulley  the  resistance  R  {Fig.  114)  is  applied  to  the  hook. 

234.  The  conditions  of  equilibrium  in  the  fixed  pulley 
require  the  equality  of  the  power  P,  and  the  resistance  Q, 
{Fig.  113)  ;  for,  if  the  intensities  of  these  forces  were  unequal, 
the  greater  of  the  two  would  prevail. 

This  property'may  also  be  demonstrated  in  the  following 
manner  :  we  prolong  the  directions  of  the  two  forces  which 
act  tangentially,  until  they  intersect  at  the  point  E  ;  their 
resultant  will  pass  through  this  point  ;  and  since  the  effect  of 
this  resultant  is  destroyed  by  the  resistance  of  the  axis  of  the 
pulley  at  0,  the  resultant  must  likewise  pass  through  this 
point.  But  the  triangles  EP'O,  EQ,'0  being  identical,  the 
angle  P'EQ,'  is  bisected  by  the  direction  of  the  resultant; 
whence  it  follows  that  the  force  P  is  equal  in  intensity  to 
the  force  Q,. 

235.  Let  there  be  now  taken  the  equal  parts  'Eg  and  EA, 


PULLEY.  117" 

and  construct  the  parallelogram  ^gfh  ;  the  forces  P  and  Q. 
being  represented  by  the  lines  E^  and  E/i,  their  resultant  R 
will  be  represented  by  E/:  we  shall  thus  have  the  proportion 

P  :  Gl  :  R  :  :  E^  :  E/i  :  E/; 

and  from  the  similarity  of  the  triangles  E^/  and  P'OQ', 
whose  sides  are  respectively  perpendicular  to  each  other,  we 
obtain 

FO  :  Oa'  :  P'a'  :  :  E^  :  E/i  :  E/j 
hence, 

P:Q  :R::FO  :  OQ,' :  FQ': 

from  which  we  conclude,  that  in  the  fixed  pulley,  each  of 
the  forces  is  to  the  resultant,  or  the  pressure  upon  the  point 
of  support,  as  the  radius  of  the  'pulley  to  the  chord  of  the  arc 
with  ivhich  the  rope  is  in  contact. 

The  equality  of  the  forces  P  and  Q  having  been  demon- 
strated, it  follows  that  the  advantage  of  the  fixed  pulley  con- 
sists only  in  changing  the  direction  of  the  power. 

236.  Let  the  cord  QABP  {Fig.  114)  be  supposed  to  em- 
brace the  arc  AB  of  a  moveable  pulley,  one  extremity  of  the 
cord  being  attached  to  the  fixed  point  Q,  ;  and  let  a  power  P 
be  applied  to  the  other  extremity,  for  the  purpose  of  sustain- 
ing a  resistance  R.  The  reaction  exerted  by  the  fixed  point 
Q,  will  be  similar  in  its  effect  to  a  force  Q,,  and  the  conditiorus 
of  equilibrium  between  P,  Q.,  and  R  will  be  the  same  as  in  the 
case  of  the  fixed  pulley,  except  that  the  resistance  which  was 
then  denoted  by  (i,will  in  the  present  case  be  represented  by 
R.  Thus,  the  relation  between  the  power  and  resistance  will 
be  determined  from  the  proportion. 

P  :  R  :  :  radius  :  chord  of  the  arc  AB. 

As  the  intensity  of  the  power  may  be  less  than  that  of  the  re- 
sistance, the  moveable  pulley  may  effect  a  gain  of  power. 

When  the  cords  are  parallel,  the  preceding  proportion  be- 
comes 

P  :  R  :  ;  radius  :  diameter  :  :  1  :  2, 
and  the  power  is  then  equal  to  one-half  the  resistance. 

If  the  chord  of  the  arc  be  equal  to  the  radius,  the  power  and 
resistance  will  become  equal  ;  and  when  the  radius  exceeds 


118  STATICS. 

the  chord,  the  use  of  the  moveable  pulley  will  induce  a  loss  of 
power. 

237.  By  the  combination  of  a  number  of  moveable  pulleys 
we  may  succeed  in  raising  enormous  weights  by  the  applica- 
tion of  a  very  small  force  :  the  pulleys  may  be  arranged  in  the 
following  manner  : 

The  weight  R  {Fig.  115)  is  suspended  from  the  hook  of  the 
moveable  pulley  ABD,  around  which  a  cord  is  passed  having 
one  of  its  extremities  attached  to  the  fixed  point  K,  and 
the  othej:  to  the  hook  of  the  pulley  A'B'D'.  This  second 
pulley  is  in  like  manner  supported  by  a  cord,  attached  at  one 
end  to  the  point  K',  and  at  the  other  to  the  hook  of  the  pulley 
A"B"D"  ;  and  the  same  arrangement  is  continued  to  the  last 
pulley,  which  is  embraced  by  a  cord  connected  at  one  end  with 
a  fixed  point  K",  the  force  P  being  applied  to  the  other.  If 
an  equilibrium  subsists  throughout  the  system,  the  tensions 
of  the  cords  AE,  A'E',  (fcc.  being  denoted  by  T,  T',  <fcc.,  we 
shall  have,  by  supposing  there  are  three  pulleys, 

R  :  T  :  :  AB  :  AC, 

T  :  T'  :  :  A'B'  :  AC, 

T'  :  P  :  :  A"B"  :  A"C". 

These  proportions  being  multiplied  together  give 

R  :  P  :  :  AB  X  A'B'  x  A"B"  :  AC  X  A'C'x  A"C"  ; 

from  which  we  conclude,  that  the  power  is  to  the  resistance 
as  the  continued  product  of  the  radii  of  the  pulleys  is  to  the 
continued  product  of  the  chords  of  the  arcs  embraced  by  the 
ropes. 

When  the  ropes  are  parallel  these  chords  become  diameters, 
and  the  proportion  is  reduced  to 

R  :  P  :  :  2=»  :  1  ; 

and,  in  general,  for  a  number  of  pulleys  denoted  by  n, 
R  :  P  :  :  2"  :  1. 

238,  This  arrangement  of  pulleys  is  seldom  adopted,  on 
account  of  its  requiring  too  great  a  space.  For,  if  the  ropes 
be  parallel,  as  represented  in  Pig.  116,  and  the  centre  of  the 
pulley  BOC  be  raised  through  a  height  denoted  by  h,  the 
line  BC  being  brought  into  the  position  he,  each  branch  of  the 


PULLEY.  11^ 

cord  D"CBX  must  be  shortened  by  the  quantity  Bb=Cc=h, 
and  the  whole  rope  will  therefore  be  shortened  by  the  quan- 
tity 2h  :  consequently,  the  pulley  AE  will  rise  through  the 
distance  2h  ;  for  a  similar  reason,  the  third  pulley  will  rise 
through  a  distance  Ah,  equal  to  twice  that  described  by  the 
second  ;  the  same  may  be  said  of  any  number  of  pulleys  : 
and  the  power  P  applied  to  the  extremity  of  the  last  rope 
must  rise  through  twice  the  distance  which  the  last  pulley 
ascends.  Thus  with  a  number  of  pulleys  represented  by  n, 
the  power  will  rise  through  a  distance  expressed  by  2"h,  and 
we  therefore  lose  in  the  space  described,  in  the  same  propor- 
tion that  we  gain  in  power. 

To  estimate  the  pressures  sustained  by  the  fixed  points  D, 
D',  D",  (fee,  we  will  represent  them  by  Q.,  Q.',  Q,";  then,  calling 
S  and  X  the  tensions  of  the  cords  SA  and  XB,  we  shall  have 

p=Q,  s=a',  x=a''; 

which  values  substituted  in  the  proportions 
.    P  :  S  :  :  1  :  2, 
S  :  X  :  :  1  ;  2, 
give 

a-2P,  a"=4P. 

239.  The  nviiffie  is  a  combination  of  several  pulleys,  all  of 
which  are  disposed  in  the  same  block,  and  have  a  common 
cord  passing  around  their  respective  circumferences. 

To  determine  the  relation  between  the  power  and  the 
resistance  in  the  muffle,  represented  in  Fig.  117,  we  remark 
that  the  several  branches  of  the  rope  must  be  equally  stretched, 
and  that  these  tensions  acting  conjointly  must  produce 
an  equilibrium  with  the  resistance  R,  which  may  therefore 
be  regarded  as  solicited  by  six  equal  and  parallel  forces.  The 
force  d  will  be  measured  by  the  intensity  of  one  of  these 
equal  forces,  and  will  consequently  be  equal  to  one-sixth  of 
the  resistance.  Or,  in  general,  the  power  will  he  to  the  resist- 
ance as  unity  to  the  number  of  cords  which  support  the  resist- 
ance. 

240.  In  the  use  of  either  system  of  pulleys,  a  certain  force 
will  be  necessary  to  overcome  the  weights  of  the  moveable 
pulleys.     The  value  of  this  force  may  be  readily  estimated 


120  STATICS. 

by  regarding  the  weight  of  each  pulley  as  an  additional  force 
applied  to  its  hook.  Thus,  in  the  system  with  separate  ropes 
represented  in  Pig,  116,  the  weight  of  the  pulley  BOC  may 
be  considered  as  applied  to  the  hook,  and  will  be  equally  sup- 
ported by  the  cords  BX  and  CD"  :  and  since  the  addition  of 
every  moveable  pulley  reduces  the  power  one-half,  it  follows, 
that  the  power  will  support  one-half  the  weight  of  the  upper 
pulley,  one-fourth  of  the  weight  of  AE,  and  one-eighth  of  the 
weight  of  BC.  In  the  muffle  {Fig.  IIT'),  the  weight  of  the 
moveable  block  being  equally  distributed  among  the  cords, 
the  power  will  sustain  one-sixth  of  this  weight. 

Of  the  Wheel  and  Axle. 

241.  This  machine  is  composed  of  a  wheel  firmly  con- 
nected with  a  cylindrical  axis.  To  the  circumference  of  the 
wheel  a  cord  is  attached,  by  means  of  which  we  can  impart 
to  it  a  motion  of  rotation,  the  effect  of  which  is  immediately 
communicated  to  the  cylinder  ;  a  second  cord  being  wrapped 
around  the  cylinder  in  a  contrary  direction,  communicates 
motion  to  the  resistance  which  is  to  be  overcome.  The  axis 
is  supported  at  its  extremities  by  two  cylindrical  pivots  which 
are  of  less  dfômeter  than  the  cylinder  itself,  and  permit  it  to 
turn  freely  about  the  points  of  support. 

242.  To  investigate  the  relation  between  the  power  and 
resistance  in  this  machine,  let  us  suppose  its  axis  AB  (Fig. 
118)  to  have  a  horizontal  position,  and  let  a  horizontal  plane 
be  drawn  through  this  axis,  intersecting  the  direction  of  the 
power  P  at  the  point  F.  Represent  the  intensity  of  the  force 
P  by  the  portion  FP  of  its  line  of  direction,  and  decompose  it 
into  two  forces  FL=P'  acting  in  a  horizontal  direction,  and 
FK=P"  acting  in  a  vertical  direction.  The  direction  of  the 
force  P'  being  prolonged  will  intersect  the  fixed  axis,  and  the 
effect  of  this  force  will  be  destroyed  by  the  reaction  of  the  axis. 

If  motion  be  communicated  by  the  force  P,  the  point  of 
application  F  of  the  vertical  component  P"  will  descend,  and 
the  resistance  R  will  ascend,  while  the  point  M,  the  intersec- 
tion of  the  line  HF  with  the  axis  of  the  cylinder,  will  remain 
immoveable.    The  point  M  may  therefore  be  regarded  as  the 


WHEEL    AND    AXLE.  121 

fulcrum  of  a  lever  HF,  to  the  extremities  of  which  the  forces 
R  and  P"  are  applied  ;  we  shall  consequently  have,  by  the 
property  of  the  lever,  when  an  equilibrium  subsists, 

P"  :  R  :  :  MH  :  MF. 
Again,  the  planes  of  the  wheel  and  of  the  section  EOH  being 
perpendicular  to  the  axis  of  the  cylinder,  the  triangles  HIM 
MCF  are  right-angled  and  similar  :  hence, 

MH  :  MF  :  :  HI  :  CF. 
From  these  proportions  we  deduce 

P"  :  R  :  :  HI  :  CF. 
Let  (p  represent  the  angle  FPK  {Fig.  118  and  119),  we  shall 
have 

FPK=DFC=.p, 
and  consequently, 

FK=FPxsin^,    DC=CFxsin^; 

P"=P  sin  <p,    CF=4—  ; 

sm  <f 

these  values  bdng  substituted  in  the  preceding  proportion, 
give 

Pxsin*:R::HI:-Ç^;       ^"^ 

sni  <p 

whence, 

PxDC=RxHI: 

and  from  this  we  deduce  the  following  proportion, 

P  :  R  :  :  HI  :  DC (122). 

It  thus  appears  that  the  conditions  of  equilibrium  in  the 
wheel  and  axle  require  that  the  -power  shall  he  to  the  resist' 
mice  as  the  radius  of  the  cylinder  to  that  of  the  loheel. 

243.  The  pressures  sustained  by  the  pivots  A  and  B  arise 
from  three  distinct  causes,  viz.  :  the  power,  the  resistance, 
and  the  weight  of  the  machine.  If  T  represent  the  value  of 
this  weight,  the  centre  of  gravity  of  the  machine  being  situ- 
ated at  the  point  G,  we  may  regard  the  weight  T  as  sus- 
pended from  the  point  G  :  the  machine  being  symmetrical 
with  r€spect  to  its  axis,  this  point  will  be  situated  upon  the 
axis.     Then,  if  the  power  P  be  replaced  by  its  components 


122 


STATICS. 


P'  and  P",  it  will  be  simply  necessary  to  substitute  for  the 
four  forces  F,  P",  R,  and  T,  two  others  applied  at  A  and  B 
respectively. 

The  forces  R  and  T  having  been  determined  by  experi- 
ment, P'  and  P"  may  be  expressed  in  functions  of  R.  For, 
we  have  {Pig.  118  and  119) 

P'=FL=P  cos  FPK,     P"=FK=P  sin  FPK  ; 
or, 

P'=P  cos  <p,    P"=P  sin  ^ (123). 

But  the  angle  p  being  equal  to  the  angle  CFD,  we  obtain 

1  :  cos  <5  :  :  CF  :  DF,     1  :  sin  <5  :  :  CF  :  CD  ; 
whence, 

DF  CD 

cosp=_,    sinç=-. 

Substituting  these  values  in  equations(123),  there  results 

CF'  CF' 

and  replacing  P  by  its  value  given  in  the  proportion  (122), 
we  obtain 

p,    R.HI.DF  Rjn 

""   DC.CF  '        ""  CF  • 
The  vertical  forces  R  and  P"  being  regarded  as  acting  at  the 
extremities  of  a  lever  whose  fulcrum  is  situated  at  the  point 
M,  their  resultant  will  pass  through  this  point,  and  its  value 
will  be  expressed  by  R+P". 

If  Z  and  Z'  denote  the  effects  produced  by  this  resultant 
upon  the  points  A  and  B,  their  values  will  be  determined  by 
the  proportions 

AB  :  BM  :  :  R+P"  :  Z, 

AB  :  AM  :  :  R+P  "  :  Z'. 

Representing  in  like  manner  by  U  and  U',  the  components  of 

T  acting  on  the  points  of  support,  we  shall  have 

AB  :  BG  :  :  T  :  U, 

AB  :  AG  :  :  T  :  U'. 

The  forces  U  and  U'  being  vertical,  they  must  be  added  to 

Z  and  Z'  respectively.     The  horizontal  force  P',  which  acts 

at  C,  the  centre  of  the  wheel,  being  likewise  decomposed  into 

two  components  Y  and  Y'  applied  at  the  points  A  and  B,  the 


TVHEEL    AND   AXLE.  123 

values  of  these  components  Y  and  Y'  will  result  from  the 
proportions 

AB  :  CB  :  :  F  :  Y, 

AB  :  AC  :  :  F  :  Y'. 
Thus,  having  constructed  two  rectangles,  the  first  of  which 
shall  have  a  height  Z  +  U  and  a  base  Y,  and  the  second  a 
height  Z'+U'  and  a  base  Y',  the  diagonals  of  these  rectangles 
will  represent  the  pressures  on  the  points  of  support  ;  and  the 
angles  formed  by  the  diagonals  with  the  sides  of  the  rectangles 
will  make  known  the  directions  in  which  these  pressures  are 
exerted. 

244.  If  regard  be  had  to  the  thickness  of  the  cords,  we  must 
consider  the  effects  of  the  powers  as  transmitted  through  the 
axes  of  the  cords  ;  thus,  the  radius  of  the  cylinder  and  that 
of  the  wheel  must  be  increased  by  the  semi-diameter  of  the 
cord,  and  we  shall  then  have  the  proportion  :  the  power  is  to 
the  resistance  as  the  sum  of  the  radii  of  the  cylinder  and 
cord  to  the  sum  of  the  radii  of  the  wheel  and  cord. 

245.  The  capstan  is  a  variety  of  the  wheel  and  axle,  in 
which  the  axis  of  the  cylinder  has  a  vertical  position. 

246.  Let  it  now  be  supposed  that  we  have  a  system  of 
wheels  and  axles  arranged  in  the  following  order  : 

The  power  P  applied  to  the  circumference  of  the  wheel 
AD  {Pig.  120)  communicates  motion  to  the  cylinder  BC, 
from  which  the  motion  is  transmitted  to  a  second  wheel 
A'D',  by  means  of  the  cord  BA'.  The  wheel  A'D'  turns  the 
axle  OB',  to  which  is  attached  the  cord  B'A",  and  a  similar 
arranarement  is  continued  to  the  last  axle,  from  which  the 
resistance  R  is  suspended. 

When  the  system  is  in  equilibrio,  if  we  denote  by  T,  T',  T", 
&c.,  the  tensions  of  the  cords  BA',  B'A",  &c.,  we  shall  have 
For  the  first  wheel  and  axle,     P   :  T  :  :  OB     :  OA, 
For  the  second       .     .     .     .     T  :  T'  :  :  O'B'  :  OA', 

For  the  third T'  :  R  :  :  0"B"  :  0"A". 

These  proportions  being  multiplied  together,  there  results 
P  ;  R  :  :  OB X OB' X 0"B"  :  O A x O'A' x O "A" ; 

whence, 

P     OBxO'B'xO"B" 


R     OAxO'A'xO'A 


7/ J 


124 


STATICS. 


irom  which  we  conclude  that  the  -power  is  'o  the  resistance 
as  the  continued  product  of  the  radii  of  the  axles  to  the  con- 
tinued product  of  the  radii  of  the  ivheels. 

If  the  radius  of  each  axle  be  supposed  equal  to  the  7i"  part 
of  the  radius  of  its  wheel,  the  preceding  proportion  will 
become 

p  :  R  :  :  2^x  — X^^'  ":  OA X O'A' X 0"A", 
)i  n  2i 

which  reduces  to 

P  :  R  :  :  1  : n\ 

247.  The  different  parts  of  a  system  of  wheel-work  are 
frequently  caused  to  act  upon  each  other  by  means  of  teeth 
projecting  from  the  several  circumferences.  These  teeth 
perform  the  same  office  as  the  cords  in  Pig.  ISO.  Each 
toothed-wheel  is  traversed  by  an  axis  bearing  a  smaller  wheel 
which  is  called  a  pinion^  and  the  teeth  of  this  pinion  are 
called  leaves.  The  first  wheel  turns  its  own  pinion,  both 
being  firmly  connected  with  the  same  axis,  and  the  leaves  of 
the  pinion  catching  into  the  teeth  of  the  second  wheel,  com- 
municate a  motion  to  it  in  a  direction  contrary  to  that  o  f  the 
first  wheel,  hi  a  similar  manner,  the  pinion  of  the  second 
wheel  transmits  a  motion  to  the  third  wheel,  and  the  same 
arrangement  is  continued  throughout  the  system.  The 
pinions  replace  the  axles  of  the  preceding  combination,  and 
hence  the  condition  of  equilibrium  is,  that  the  power  shall  be 
to  the  resistance  as  the  continued  product  of  the  radii  of  the 
2nnions  to  the  conti/tued  jjroduct  of  the  radii  of  the  wheels. 

248.  Let  D,  D',  D",  &c.  represent  the  numbers  of  teeth  in 
the  wheels  A,  A',  A",  &c.  [Pig.  121),  and  d,  d',  d",  <fcc.  the 
numbers  of  leaves  in  the  pinions  a,  a',  a",  &c.  ;  and  let  us 
suppose  that  while  the  wheel  A  makes  N  turns,  the  wheels 
A',  A",  (fcc.  make  respectively  N',  N",  &c.  turns.  At  each 
revolution  of  the  wheel  A,  the  pinion  a  will  engage  in  suc- 
cession all  its  leaves  in  the  teeth  of  the  wheel  A'  ;  so  that  in 
N  revolutions  it  will  engage  with  A',  a  number  of  teeth  ex- 
pressed by  Nrf  :  in  like  manner,  the  wheel  A'  making  N'  turns 
must  engage  with  the  pinion  a,  a  number  of  teeth  expressed 
by  N'D'j  and  since  the  numbers  of  teeth  and  leaves  which  the 


WHEEL    AND   AXLE.  125 

wheel  A'  and  the  pinion  a  mutually  interlock  are  equal  to 
each  other,  we  must  necessarily  have 

for  a  similar  reason,  the  other  wheels  will  furnish  the 
equations 

N"D"=N'c;',     N"'D"'=N"<i",  (fee. 
These  equations  being  multiplied  together,  there  results 

N"'D'D"D"'=Ndd'd"', 
whence, 

D'D'D"* 

For  example,  if  it  were  required  to  determine  the  number  of 
teeth  which  should  be  employed  in  order  that  the  wheel  A'" 
should  make  one  revolution  while  the  wheel  A  performs  60, 
we  should  have 

N"'  =  l,     N=60,     l=60^j^ (124). 

The  numbers  d,  d'  and  d"  being  assumed  arbitrarily,  we  will 
suppose  tZ=4,  d'=5,  d"  =7  ;  this  supposition  will  reduce  the 
last  of  the  equations  (124)  to 

D'xD"xD"'=60x4x  5x7=8400. 
The  number  S400  being  divided  into  the  three  factors  12,  25, 
and  28,  will  evidently  furnish  a  solution  to  the  problem,  since 
the  quantities  D',  D",  and  D'"  may  be  made  respectively 
equal  to  these  factors.  The  problem  obviously  admits  of  an 
indefinite  number  of  solutions. 

The  quantity  N'"  must  be  assumed  less  than  N,  since  we 
have  supposed  d<D',  d'<D",  d"<D"',  and  the  wheel  A'"  will 
therefore  make  a  less  number  of  revolutions  in  a  given  time 
than  the  wheel  A. 

249.  The  theory  of  the  jack-screw  is  likewise  to  be  referred 
to  that  of  the  wheel  and  axle.  There  are  two  varieties  of 
this  machine,  the  simple  and  the  compound.  The  simple  jack 
is  composed  of  a  toothed  bar  of  iron  AB  {Pig.  122)  which 
slides  in  a  case  CD.  The  teeth  of  this  bar  work  in  the  leaves 
of  the  pinion  EF,  which  is  put  in  motion  by  means  of  a 
crank  G  ;  thus,  the  teeth  of  the  bar  being  subjected  to  a 
pressure  from  the  leaves  of  the  pinion,  the  bar  will  move  in 


136  STATICS. 

the  direction  of  its  length,  and  will  overcome  a  resistance  at 
A.  In  this  machine,  the  crank  and  pinion  perform  the 
offices  of  the  wheel  and  the  axle  in  the  common  machine,  and 
the  conditions  of  equilibrium  may  therefore  be  stated  thus  : 
the  power  is  to  the  resistance  as  the  radius  of  the  pinion  to  the 
radius  of  the  crank. 

250.  In  the  compound  jack-screw,  the  motion  is  cormnu- 
nicated  by  means  of  a  crank  to  a  pinion,  the  leaves  of  which 
work  into  the  teeth  of  a  wheel  ;  the  axis  of  this  wheel  carries 
a  second  pinion,  which  in  its  turn  communicates  motion  to 
a  second  wheel,  and  the  same  arrangement  is  continued  to 
the  last  pinion,  whose  leaves  act  on  the  teeth  of  the  iron  bar. 

The  condition  of  equilibrium  in  this  machine  obviously  is, 
that  the  power  shall  he  to  the  resistance  as  the  continued  pro- 
duct of  the  radii  of  the  pinions  to  the  continued  product  of 
tht  radii  of  tlie  wheels  and  the  radius  of  the  crank. 

Of  the  Inclined  Plane. 

2b\.  This  machine  consists  of  a  plane  inclined  to  the 
horizon  :  its  object  is  to  support  in  part  the  weight  of  a  body 
placed  upon  it. 

Let  M  represent  a  body  {Pig.  123)  the  weight  of  which  is 
supposed  concentrated  at  its  centre  of  gravity,  and  exerted  in 
the  vertical  direction  MP.  In  order  that  this  body  may  be 
sustained  in  equilibrio  upon  the  inclined  plane  by  the  appli- 
cation of  a  force  Q,,  it  is  necessary  that  this  force  Q,  and  the 
weight  of  the  body  represented  by  P,  should  have  a  single 
resultant  ;  this  condition  can  only  be  fulfilled  when,  the 
directions  of  the  forces  intersect  at  some  point  M  :  but  the 
hne  MP  being  vertical,  and  passing  through  the  centre  of 
gravity,  the  plane  of  the  forces  PMQ,  must  likewise  be  ver- 
tical, and  must  contain  the  centre  of  gravity.  Thus  the  first 
condition  of  equilibrium  requires  that  the  direction  of  the  re- 
sultant be  situated  in  a  vertical  plane  passing  through  the 
centre  of  gravity  of  the  body.  The  second  condition  is,  that 
the  resultant  MN  of  the  two  forces  P  andQ,  shall  be  destroyed 
by  the  resistance  of  the-inclined  plane,  which  condition  can 
only  be  satisfied  when  the  direction  of  this  resultant  is  per- 


INCLINED   PLANE.  127 

pendicular  to  the  plane,  and  intersects  it  at  some  point  within 
the  polygon  formed  by  connecting  the  extreme  points  of  can- 
tact  of  the  body  and  the  plane. 

252.  The  preceding  conditions  being  fulfilled,  we  will  sup- 
pose KL  to  represent  a  body  {Fig.  123)  retained  in  equili- 
brio  upon  an  inclined  plane  by  the  application  of  a  force 
d.  Let  the  lines  ME  and  MF  be  taken  proportional  to 
the  weight  P  and  the  force  Q,,  and  let  the  parallelogram 
FMER  be  constructed  :  the  diagonal  MR  will  represent  the 
pressure  exerted  by  the  body  against  the  plane,  and  if  this 
pressure  be  denoted  by  R,  we  shall  have 

a  :  P  :  R  :  :  sin  PMR  :  sin  dMR  :  sin  PMQ, (125). 

The  triangles  APO  and  OMN  being  similar,  the  angles  PMR 
and  CAB  will  be  equal  to  each  other,  and  therefore 

sin  PMR=sin  A=-—  ; 
AO 

this  value  being  substituted  in  the  proportion  (125),  we  ob- 
tain 

a  :  P  :  R  :  :  CB  :  AC xsin  aMR  :  AC  Xsin  PMQ,. 

253.  If  the  direction  of  the  power  be  parallel  to  the  plane 
{Fig.  123),  the  triangles  MER  and  ACB  will  be  similar,  since 
the  angles  C  and  E  are  then  equal  to  each  other,  and  we  have 
the  proportion 

ER  :  ME  :  :  CB  :  AC  ; 

from  which  we  conclude  that  when  the  power  acts  parallel 
to  the  plane,  the  power  Q,  is  to  the  loeight  P  as  the  height  of 
the  plane  is  to  its  length. 

254.  When  the  power  becomes  parallel  to  the  base  of  the 
plane  {Fig.  124),  the  similar  triangles  MER  and  CAB  give 
the  proportion 

ER  :  EM  :  :  CB  :  AB, 

or, 

a  :  P  :  :  CB  :  AB  ; 

thus,  in  this  case,  the  poiver  is  to  the  weight  as  the  height  of 
the  plane  is  to  the  base. 

255.  The  angle  Abeing  supposed  equal  to  45°, and  the  power 
applied  parallel  to  the  base,  the  weight  and  power  will  be- 
come equal  ;  if  the  angle  A  be  less  than  45°,  the  weight  will 


126  STATICS. 

be  greater  than  the  power,  and  if  A  be  greater  than  45°,  the 
power  will  exceed  the  weight,  or  the  use  of  the  machine  will 

occasion  a  loss  of  power. 

256.  If  a  body  be  sustained  in  equilibrio  between  two  in- 
clined planes,  the  conditions  of  equilibrium  will  require  that 
the  weight  of  the  body  be  susceptible  of  being  resolved  into 
two  components  which  shall  be  respectively  perpendicular  to 
these  planes,  and  shall  intersect  them  at  points  situated  within 
the  polygons  formed  by  joining  the  points  of  contact  of  the 
body  with  each  plane.  The  line  of  direction  of  the  weight 
being  vertical,  the  plane  of  its  components  will  likewise  be 
vertical  :  and  since  these  components  are  respectively  per- 
pendicular to  the  inclined  planes,  their  plane  will  be  perpen- 
dicular to  the  common  intersection  of  the  inclined  planes  : 
hence,  this  intersection  must  be  a  horizontal  line. 

The  pressures  sustained  by  these  planes  may  be  readily 
determined  by  constructing  the  parallelogram  of  forces,  whose 
diagonal  shall  represent  the  weight  of  the  body,  and  whose 
sides  shall  be  perpendicular  to  the  inclined  planes. 

Of  the  Screio. 

257.  Let  the  sides  of  the  rectangle  AM'  {Fig.  125)  be 
divided  into  equal  parts  by  the  parallel  lines  BB',  CC,  (fee, 
and  let  the  diagonals  AB',  BC,  &c.  be  drawn.  If  the  rectan- 
gle M'A  be  then  applied  to  the  surface  of  a  right  cylinder 
with  a  circulai  base,  the  circumference  of  which  is  equal  to 
the  line  AA',  in  such  manner  that  the  right  lines  MA  and  M'A' 
shall  be  caused  to  coincide,  the  points  A,  B,  <fcc.  will  fall  upon 
the  points  A',  B',  (fee.  respectively,  and  the  diagonals  will  trace 
upon  the  surface  of  the  cylinder  PQNM  {Fig.  126)  a  curve 
PRSTUV  (fee,  which  is  called  a  helix. 

258.  The  characteristic  property  of  this  curve  is  that  the 
tangent  at  every  point  is  equally  inclined  to  the  element  of 
the  cylinder  passing  through  that  point  :  this  is  obvious  from 
the  manner  in  which  the  curve  is  generated. 

The  distances  mn,  7?i'n',  m"n",  (fee.  {Fig.  125)  being  equal, 
their  equality  will  be  preserved  when  the  rectangle  is  applied 
to  the  surface  of  the  cylinder  :  consequently,  if  we  assume 


SCREW.  129 

mn  as  the  base  of  an  isosceles  triangle  mno,  the  plane  of  which 
passes  through  the  axis  of  the  cyhnder,  and  cause  the  triangle 
to  move  around  the  cylinder,  in  such  manner  that  the  points 
9n  and  7t  shall  constantly  remain  on  two  adjacent  helices,  the 
plane  of  the  triangle  continuing  to  pass  through  the  axis  of 
the  cylinder,  there  will  be  generated  by  this  motion  a  project- 
ing fillet  which  will  completely  envelop  the  cylinder  MQ,. 
The  cylinder  and  fillet  taken  conjointly  constitute  the  screw, 
and  the  latter  is  usually  called  the  thread  of  the  screw.  This 
thread  is  sometimes  generated  by  the  motion  of  a  rectangle, 
instead  of  a  triangle. 

259.  The  nut  is  composed  of  a  hollow  piece,  having  a 
spiral  groove  cut  in  its  interior,  in  which  the  threads  of  the 
screw  work.  It  may  be  regarded  as  forming  the  mould  of  a 
portion  of  the  screw. 

The  screw  can  be  readily  turned  within  the  nut,  and  at 
each  revolution  passes  over  a  distance  in  the  direction  of  its 
length  equal  to  the  distance  between  the  threads. 

Since  the  conditions  of  the  problem  are  precisely  the  same, 
whether  we  regard  the  nut  as  turning  on  the  screw,  or  the 
screw  as  turning  within  the  nut,  we  will  adopt  the  first 
hypothesis. 

260.  To  determine  the  conditions  of  equilibrium  in  this 
machine,  we  will  suppose  the  nut  to  be  placed  on  its  screw, 
and  the  axis  of  the  screw  to  have  a  vertical  position.  Let 
the  nut  be  divided  into  any  number  of  particles,  whose  weights 
are  denoted  by  m,  m',  m",  &.C.,  each  of  which  rests  on  some 
point  of  the  screw  ;  and  let  us  determine  the  force  necessary 
to  sustain  any  one  particle  m  {Mg.  127). 

The  particle  m,  being  connected  with  the  axis  of  the  screw 
in  such  manner  that  its  distance  from  the  axis  shall  remain 
invariable,  must,  if  unsupported,  descend  along  a  helix,  every 
point  of  which  will  be  at  the  distance  mC  from  the  axis. 
Thus,  by  regarding  this  helix  as  an  inclined  plane,  the  height 
of  this  plane  will  be  the  distance  between  the  threads,  and 
its  base  will  be  the  circumference  described  with  mC  as  a 
radius. 

Let  us  suppose  a  horizontal  force  P  {Pig:  128)  to  be  ap- 
plied immediately  to  the  particle  w,  for  the  purpose  of 

I 


130  STATICS. 

sustaining  it  in  equilibrio  upon  the  inclined  plane.  By  con- 
structing the  right-angled  triangle  KHw,  whose  height  shall 
be  the  distance  between  the  threads,  and  its  base  the  circum- 
ference described  with  the  radius  mC,  we  shall  obtain  by  the 
principle  of  the  inclined  plane  (Art,  254), 

P  :  m  :  :  height  :  KH  ; 
or, 

P  :  m  :  ;  mH  :  circumference  Cm (126). 

But  if  the  point  of  application  of  the  power  be  transferred 
from  the  point  m  to  the  point  D,  the  extremity  of  the  lever 
CD,  the  force  d,  which  applied  at  this  point  will  produce  the 
same  effect  as  the  force  P  applied  at  m,  can  be  determined 
from  the  following  proportion, 

a  :  P  :  :  Cw  :  CD  ; 
or, 

Q,  :  P  :  :  circumference  Cm  :  circumference  CD. 
And  by  comparing  this  proportion  with  (126),  we  obtain 

Q,  :  m  :  :  mH  :  circumference  CD. 
Thus,  for  the  particle  m,  the  power  is  to  the  weight  as  the 
distance  between  the  threads  is  to  the  circumference  described 
by  the  power. 

This  proportion  being  true,  whatever  may  be  the  distance 
of  the  particle  m  from  the  axis  of  the  cylinder^  we  shall  ob- 
tain for  the  other  points  in  the  surface  of  the  screw,  which 
support  the  weights  m,  m",  (kc,  by  means  of  the  ^oroes 
Q,',  Q.",  (fcc,  applied  at  the  same  distance  CD, 

:  :  mH  :  circumference  CD, 
:  :  mH  :  circumference  CD, 
:  :  mH  :  circumference  CD, 
(kc.        &c.        &.C. 
From  these  proportions  and  the  preceding,  we  deduce 

„_  771  XmH  ^,_  m'XmH  ^„_  7;i"XmH 
"circumf  CD'  circumf  CD'  ""circumf  CD'  '  ^  ^' 
These  values  are  independent  of  the  distances  of  the  points 
m,  m',  m",  ôcc,  from  the  axis  of  the  cylinder  ;  and  since  the 
forces  Q,,  CI',  Q,",  &.c,  were  supposed  applied  at  equal  dis- 
tances from  the  axis,  they  will  communicate  to  the  nut  the 


a 

:  m 

a" 

:  m' 

a" 

:m' 

WEDGE.  131 

same  motion  of  rotation  as  would  be  imparted  by  a  single 
force  equal  to  their  sum,  and  acting  along  the  line  DQ,. 
Thus,  by  adding  the  equations  (127),  we  find 

circumf.  CD 


(w+m'+m"+&c.)=(Cl+Q,'+a"+&c.)- 


mH 


and  since  the  sum  (m+y'i'+w"+&c.)  represents  the  entire 
weight  M  of  the  nut,  we  shall  have,  after  replacing  the  sum  of 
the  forces  Q,  Q,',  Gl",  (fee,  by  a  single  force  Q,; , 

jj^         circumfCD- 
ma. 
whence, 

Q.y  :  M  :  :  mH  :  circumference  CD  : 
or,  the  power  is  to  the  toeight  as  the  distance  between  the 
threads  is  to  the  circumference  described  by  the  power. 

It  thus  appears  that  the  machine  will  be  rendered  more 
powerful  by  applying  the  force  at  a  greater  distance  from  the 
axis,  or  by  diminishing  the  distance  between  the  threads 
of  the  screw. 

Of  the  Wedge. 

261.  The  wedge  is  a  triangular  prism,  one  of  whose  edges 
is  introduced  into  the  crevice  of  a  body,  for  the  purpose  of 
enlarging  the  opening. 

All  cutting  instruments,  such  as  knives,  scissors,  razors, 
&.C.,  may  be  regarded  as  wedges. 

262.  The  power  is  usually  applied  by  communicating  an 
impulse  to  the  back  of  the  wedge,  in  a  direction  perpendicular 
to  it  :  if  the  direction  of  this  impulse  be  oblique,  it  may 
always  be  resolved  into  two  components,  of  which  one  shall 
be  perpendicular  to  the  back  of  the  wedge,  and  the  other  shall 
coincide  with  it.  The  first  will  produce  its  entire  effect,  the 
second  will  only  tend  to  move  the  point  of  application  of  the 
power  along  the  back  of  the  wedge. 

Let  ABC  {Fig.  129)  represent  a  profile  of  the  wedge; 
AC  and  BC  are  sections  of  its  faces,  and  AB  a  section  of  its 
back,  upon  which  the  power  is  applied  in  a  perpendicular 
direction. 

12 


132 


STATICS. 


To  determine  the  relation  between  the  power  appUed  to  the 
back  of  the  wedge  and  the  pressures  exerted  at  the  faces,  we 
will  suppose  the  power  F  to  be  represented  by  the  line  DE, 
and  draw  DM  and  DN  perpendicular  to  the  faces  AC  and  BC  : 
then,  by  constructing  the  parallelogram  DIEK,  the  compo- 
nents DI  and  DK  will  represent  the  pressures  exerted  against 
AC  and  BC.  Denoting  these  pressures  by  X  and  Y,  the 
similar  triangles  ABC  and  IDE  give  the  proportion 

DE  :  DI  :  IE  :  :  AB  :  AC  :  BC  ; 
or, 

F  :  X  :  Y  :  :  AB  :  AC  ;  BC  ; 
and  by  multiplying  the  three  last  terms  in  this  proportion  by 
the  line  GH  {Fig.  130),  we  have 

F  :  X  :  Y  :  :  AB X GH  :  AC X GH  :  BC XGH. 
The  products  AB  X  GH,  AC  X  GH,  and  BC  X  GH  express  the 
surfaces  of  the  back  and  faces  of  the  wedge,  and  we  therefore 
conclude  that  in  this  machine,  the  poiver  F  applied  to  the 
back,  and  the  efforts  X  and  Y  exerted  by  the  sides,  are  respect- 
ively propoî'tio?ial  to  the  surfaces  of  the  back  and  sides  of 
the  wedge. 

The  power  of  the  wedge  will  evidently  be  augmented 
either  by  decreasing  the  back  of  the  wedge,  or  by  increasing 
the  lengths  of  its  faces. 

Friction. 

263.  If  a  body  be  placed  upon  a  horizontal  plane,  the  action 
of  gravity  exerted  upon  it  will  be  entirely  counteracted  by 
the  resistance  of  the  plane,  and  the  least  possible  impulse 
will  communicate  a  motion  to  the  body,  if  it  be  not  retained 
by  physical  causes  which  oppose  motion.  The  most  efficient 
of  these  causes  is  friction.  This  term  is  applied  to  the  force 
which  tends  to  prevent  a  body  from  sliding  along  the  surface 
of  a  second  body,  and  which  arises  from  the  slight  inequalities 
in  the  two  surfaces  ;  the  projecting  points  of  one  surface  en- 
tering the  cavities  of  the  second  give  rise  to  a  passive  force 
which  tends  to  assist  or  oppose  the  power,  according  as  this 
power  is  employed  to  sustain  or  move  the  body. 


FRICTION.  133 

The  eiFect  of  friction  is  found  to  be  sensibly  proporiional 
to  the  pressure,  so  long  as  this  pressure  is  retained  within 
moderate  limits.  Thus,  if  we  denote  by  /  the  friction  ex- 
erted by  a  homogeneous  body  AB  {Pig.  131),  the  weight  of 
which  is  equal  to  unity,  and  if  AB'  be  supposed  equal  to  twice 
AB,  the  corresponding  friction  will  be  expressed  by  2/";  if 
AB"  be  triple  AB,  the  friction  will  be  equal  to  3/,  &.C.;  so  that 
if  F  denote  the  friction  exerted  by  the  body  AM,  which  con- 
tains a  number  N  of  units  of  weight,  we  shall  have 

F=N/- (128). 

264.  The  friction  may  be  measured  in  the  following 
manner  : 

Let  AB  (Fig.  132)  represent  the  body  which  exerts  by  its 
weight  the  unit  of  pressure  on  a  horizontal  plane  LK.  To 
the  body  is  attached  a  thread  CDE,  which  passes  over  a  fixed 
pulley,  and  sustains  the  weight  M  :  this  weight  being  grad- 
ually increased,  its  intensity  at  the  moment  when  it  is  about 
to  overcome  the  resistance  which  the  body  opposes  to  motion, 
will  measure  the  friction /,  corresponding  to  the  unit  of  pres- 
sure. 

265.  There  is  another  method  of  measuring  the  friction, 
which  results  from  the  following  theorem  :  If  a  body  MN  he 
placed  upon  an  inclined  plane  AC  {Fig.  133),  a7id  if  the 
angle  A  wliich  this  j)lane  forms  with  the  horizon  he  grad- 
ually augmented  until  the  body  is  about  to  commence  sliding 
upon  the  plane,  the  nmnerical  value  of  the  unit  of  friction 
toill  then  be  equal  to  the  tangent  of  the  angle  ichich  the 
inclined  -plane  forms  with  the  horizon. 

To  demonstrate  this  fact,  let  the  lines  GD  and  GK  be 
drawn,  respectively  perpendicular  to  AB  and  AC  ;  the  centre 
of  gravity  of  the  body  being  supposed  situated  at  the  point 
G.  Represent  by  GD  the  weight  of  the  body,  and  decom- 
pose GD  into  two  forces  GH  and  GK,  parallel  and  perpen- 
dicular to  the  inclined  plane  :  we  shall  then  have 

GH=DK=GD  sin  DGK, 
GK=GD  cos  DGK  ; 
12 


134 


STATICS. 


but  the  angles  DGK  and  CAB  are  equal  to  each  other  ;  and 
hence,  the  preceding  equations  may  be  written  thus, 
GH=GDxsinA, 
GK=GDxcos  A; 
or  if  N  expresses  the  weight  of  the  body, 
GH=NsinA, 
GK=N  cos  A. 
The  pressure  sustained  by  the  inclined  plane  being  expressed # 
by  GK=N  cos  A,  the  corresponding  friction  will  be  expressed 
by  N  cos  A/;  but  since  the  effect  of  friction  is  to  counteract 
that  tendency  which  the  body  has  to  move  along  the  plane 
when  there  is  no  friction,  it  follows,  that  an  equilibrium  will 
subsist  between  the  force  of  friction  and  the  component  of 
the  force  of  gravity,  GH=N  sin  A,  which  acts  in  the  direc- 
tion of  the  plane  ;  whence  we  obtain 

N  cos  A./=N  sin  A. 
From  this  equation  we  deduce 

/=tangA (129). 

266.  The  angle  thus  determined  is  called  the  angle  of 
friction  ;  its  value  will  remain  constant  only  when  we  adopt 
the  hypothesis  that  the  friction  varies  proportionally  to  the 
pressure.  For,  the  relation  expressed  in  (129),  has  been 
deduced  by  employing  (128),  which  expresses  this  law  ;  and 
the  law,  as  has  been  already  remarked,  exists  only  for  mode- 
rate pressures. 

267.  Since  different  substances  have  pores  of  very  unequal 
magnitudes  it  happens  that  the  friction  is  not  the  same  for  all 
bodies  ;  hence,  experiments  have  been  instituted  for  the  pur- 
pose of  determining  the  friction  peculiar  to  each. 

The  following  results  which  express  the  relation  between 
the  friction  and  the  pressure,  have  been  obtained  by  Coulomb  : 

Iron  against  iron /=0.28, 

Iron  against  brass   ....  /=0.26, 

Oak  against  oak /=0.43, 

Oak  against  fir /=0.65, 

Fir  against  fir /=0.56, 

Elm  against  elm /=0.47. 


FRICTION.  135 

These  last  results  were  obtained  when  the  friction  was 
exerted  in  the  direction  of  the  fibres  ;  but  when  the  direction 
of  the  fibres  formed  a  right  angle  with  that  of  the  motion, 
the  friction  was  found  to  be  much  less,  but  still  in  a  constant 
ratio  to  the  pressure  ;  the  results  in  this  case  were  as  follows  : 

Oak  against  fir /=0.158, 

Fir  again&t  fir /=0.167, 

Elm  against  elm /=0.100. 

It  also  appears  from  the  experiments  of  Coulomb,  that 
the  friction  exerted  by  a  body  m  motion  is  very  nearly  inde- 
pendent of  the  velocity  of  the  body. 

The  polish  of  the  body  and  the  introduction  of  an  unc- 
tuous substance  between  the  rubbing  surfaces  contribute  to 
lessen  the  effect  of  the  friction. 

268.  When  one  body  is  caused  to  roll  upon  another,  a  cer- 
tain degree  of  resistance  is  still  offered  by  friction,  but  this 
resistance  is  much  less  intense  than  in  the  case  of  a  sliding 
motion.  This  result  appears  to  be  a  consequence  of  the  dis- 
engagement of  the  inequalities  in  the  surfaces,  which  the 
motion  of  rotation  tends  to  effect. 

269.  The  general  laws  of  friction,  as  deduced  from  the 
experiments  of  Coulomb,  may  be  summed  up  as  follows  : 

1°.  Priction  varies  loith  the  polish  of  the  surface  :  Thus, 
the  resistance  opposed  by  friction  may  be  reduced  by  diminish- 
ing the  asperities  of  the  rubbing  surfaces. 

2°.  The  friction  between  bodies  of  the  same  kind  is  greater 
than  between  bodies  of  different  kinds. 

3°.  Priction  does  not  depend  on  the  extent  of  surface  iti 
contact,  the  entire  pressure  exerted  betiveen  the  bodies  remain- 
ing  the  same. 

4°.  Priction  is  proportional  to  the  pressure. 

5^.  Priction  is  diminished  by  interposing  a  substance  of 
an  unctuous  nature  betioeen  two  surfaces  which  slide  upon 
each  other. 

6°.  The  friction  is  greatly  diminished  by  substituting  a 
rolling  for  a  sliding  motion. 

270.  The  adhesion  which  takes  place  between  the  surfaces 
of  bodies  is  another  physical  cause  opposed  to  their  motion. 


136  STATICS. 

It  is  difficult  to  estimate,  in  a  precise  manner,  the  proper 
measure  of  this  effect,  in  consequence  of  its  being  hable  to  a 
very  great  increase  with  time  in  those  machines  which  are  at 
rest  ;  and,  on  the  contrary,  to  undergo  occasional  changes  in 
those  which  are  in  motion. 

The  law  which  this  force  usually  follows  is  that  of  being 
sensibly  proportional  to  the  extent  of  the  adhering  surfaces. 
Thus,  by  denoting  the  adhesion  of  a  superficial  unit  by  the 
quantity  i^,  the  adhesion  of  a  surface  whose  area  is  a  will  be 

expressed  by  a-^. 

* 

Effects  of  Pi-iction  in  certain  Machines. 

271.  Let  P  and  S  {Fig.  134)  represent  two  forces  applied 
to  a  material  point  which  rests  in  equilibrio  on  an  inclined 
plane  AB,  and  let  «  and  «'  denote  the  angles  which  the  direc- 
tions of  these  forces  make  with  the  plane.  If  we  disregard 
the  effects  of  friction  and  adhesion,  the  conditions  of  equi- 
librium will  require  the  relation 

P  cos  a=S  cos  «' (130)  ; 

but  if  friction  and  adhesion  be  considered,  since  these  two 
forces  are  opposed  to  the  motion  which  the  power  P  tends  to 
impress  in  a  direction  from  ni  towards  B,  it  will  be  necessary 
to  add  these  forces  to  the  component  of  S  in  the  direction  of 
the  plane,  which  is  expressed  by  S  cos  «'.  To  determine  their 
values,  we  remark  that  the  pressure  exerted  upon  the  inclined 
plane  is  produced  by  the  normal  components  of  the  forces 
P  and  S.  These  coniponents  are  expressed  respectively  by 
P  sin  cc  and  S  sin  «'  ;  and  their  sum  will  be  equal  to  the  entire 
pressure  which  is  denoted  by  N  in  equation  (128),  Thus,  the 
force  arising  from  friction  is  expressed  by  (P  sin  «+S  sin  x)f. 
If  we  denote  by  a  the  area  of  the  surface  in  contact  with  the 
plane,  the  adliesion  will  be  represented,  as  has  been  before 
stated,  by  the  quantity  ai'.  Consequently,  by  adding  these 
forces  to  the  second  member  of  the  equation  (130),  we  shall 
obtain  for  the  condition  of  equilibrium 

P  cosa  — S  cos*'  +  S  sin  «/+P  sin  o/'+aT//; 
from  which  we  deduce 


FRICTION.  137 

p_S  cos  et'+S/sin  ei'+a-'P  ^^l^ 

cos  a— /sin  e»  ^         '' 

272.  If,  on  the  contrary,  the  power  be  only  required  to 
retain  in  equiUbrio  the  point  m,  the  friction  and  adhesion, 
being  still  opposed  to  motion,  will  tend  to  assist  the  force  P, 
and  the  algebraic  signs  of  these  quantities  must  therefore  be 
changed.  Representing  by  P'  the  force  necessary  to  support 
in,  upon  this  hypothesis,  we  shall  have 

^,^S  cosx-Sfsmu—a-^ 

cos  <*  +/  sin  «         ^       ^' 

It  is  evident  that  the  equilibrium  may  be  preserved  by  the 
application  of  any  force  P"  in  the  direction  Pm,  provided  the 
intensity  of  this  force  be  intermediate  between  the  intensities 
P  and  P'  given  by  equations  (131)  and  (132). 

273.  The  eifect  of  friction  in  modifying  the  conditions  of 
equilibrium  in  the  lever  and  pulley  will  now  be  considered. 

Let  the  lever  be  perforated  by  a  circular  hole,  through 
which  is  passed  a  cylinder  having  a  vertical  position.  Since 
the  circumstances  will  be  the  same,  whether  we  regard  the 
lever  as  turning  about  the  cylinder,  or  the  cylinder  as  turning 
within  the  lever,  we  shall  adopt  the  first  hypothesis,  and  con- 
sider the  point  m  of  the  lever  {Mg.  135),  which,  being  in 
contact  with  the  cylinder,  is  subjected  to  the  action  of  the 
force  of  friction.  Let  the  cylinder  be  intersected  by  a  hori- 
zontal plane  passing  through  w,  and  let  this  plane  be  assumed 
as  the  co-ordinate  plane  of  x,  y.  For  the  purpose  of  simplify- 
ing the  question,  we  shall  suppose  the  resultant  R  of  all  the 
forces  applied  to  the  lever  to  be  situated  in  the  plane  of  a:,  y. 

The  intersections  of  the  cylinder  and  lever  by  the  plane  of 
.r,  y  will  be  represented  respectively  by  the  circle  mBE,  and 
the  plane  curve  GIL.  The  cylinder  being  immoveable,  the 
point  m  can  be  subject  only  to  a  circular  motion  about  the 
point  C,  at  which  the  axis  of  the  cylinder  is  intersected  by 
the  plane  of  x,  y.  If  the  point  m  remain  immoveable,  the 
equilibrium  must  result  from  the  combined  actions  of  the 
resultant  R  of  the  several  forces  applied  to  the  lever,  the  fric- 
tion, and  the  resistance  opposed  by  the  axis.  The  direction 
of  this  resistance  being  normal  to  the  surface  of  the  cylinder, 


y- 


138  STATICS. 

we  may  drop  the  consideration  of  the  fixed  cyHnder,  and  con- 
sider the  point  as  perfectly  free,  and  sustained  in  equihbrio  by 
the  three  following  forces  :  1°.  the  normal  force,  which  acts  in 
the  direction  from  C  towards  m  ;  2°.  the  friction,  which  acts 
along  the  tangent  wD  ;  3°.  the  resultant  R  of  all  the  forces 
in  the  system. 

274.  It  should  be  remarked  that  although  two  of  the  three 
forces  are  applied  at  m,  the  third  force  may  be  applied  at  any 
other  point,  provided  its  line  of  direction  passes  through  m. 

If  we  regard  the  point  of  application  of  the  third  force  as 
unknown,  the  conditions  of  equilibrium  of  the  three  forces 
will  be  expressed  by  the  equations  (52),  (53),  and  (54). 

275.  To  express  these  conditions,  we  will  suppose  the  origin 
of  co-ordinates  to  be  placed  at  C,  and  represent  by  N  the  nor- 
mal force,  which  forms  with  the  axes  angles  equal  to  «  and  /3  : 
denote  by  F  the  friction,  the  direction  of  which  forms  Avith 
the  axes  the  angles  <*'  and  ^',  and  by  h  the  radius  of  the  cylin- 
der which  is  supposed  to  be  nearly  of  the  same  size  as  the 
circular  hole  through  which  it  passes.  The  components  of 
the  force  R,  parallel  to  the  two  axes,  will  be  represented  by  X 
and  Y  respectively,  and  the  perpendicular  distance  of  this 
force  from  the  point  C  by  the  letter  r. 

This  being  premised,  the  condition  expressed  by  equation 
(52)  requires  that  the  sum  of  the  components  parallel  to  the 
axis  of  X  shall  be  equal  to  zero  ;  hence, 

N  cos  ^  +  X  +  F  cos  «'=0 (133). 

For  a  similar  reason,  the  components  parallel  to  the  axis  of 
y  give 

N  cos  /3  + Y  +  F  cos  /3'=.0 (134). 

And  the  third  equation  of  equilibrium,  which  expresses  the 
relation  between  the  moments,  gives 

Rr  +  F/i=0 (135); 

which  becomes,  by  substituting  for  F  its  value  deduced  from 
equation  (128), 

Rr  +  N/A=0 (136); 

276.  Before  employing  equations  (133)  and  (134),  it  may 
be  remarked  that  any  one  of  the  four  quantities  cos  «,  cos  «', 
cos  /8,  cos  /3',  which  appear  in  those  expressions,  will  serve  to 


:zl\ (1^^)- 


FRICTION.  139 

determine  the  remaining  three.     For,  the  angle  yCx  {Pig. 
135)  being  equal  to  a  right  angle,  we  shall  have 

cos  /3=sin  «  ; 
and  if  we  draw  the  line  FK  parallel  to  Cm  {Fig.  136),  we 
shall  obtain 

wFH=rmFK+KFH; 
or, 

consequently, 

cos  *'=cos  90°  cos  *— sin  90°  sin  *=— sin  «, 
cos  /3'=sin  «'=sin  90°  cos  «+cos  90°  sin  «=cos  «. 

By  means  of  these  values  of  cos  ji,  cos  <*',  and  cos  /3',  we  reduce 
the  equations  (133)  and  (134)  to 

N  cos  «+X— F  sin  «=0 

Nsina  +  Y+F  cos 

277.  These  equations  admit  of  a  further  reduction,  from 

the  consideration  that  the  friction  exerted  at  the  point  w  is 

proportional  to  the  normal  pressure  N  ;  thus,  by  replacing  F 

by  its  value  N/in  the  equations  (137),  we  find 

X=N/sina— N  cos» 

Y=— N/cos«— Nsin^ 

But  X  and  Y  being  rectangular  components  of  the  force  R, 

we  must  have  the  relation 

R2=X2+Y2, 
Substituting  in  this  equation  the  values  of  X  and  Y  found 
above,  we  obtain 

R2=N2(sin2  «+cos'<i)-{-N3/2(sin"  a+cos^  x), 

or, 

R2=N2(l+y2) (139). 

From  this  equation  taken  in  connexion  with  (136),  we  find 
r=±—Ih. (140). 

This  value  of  r  will  always  be  less  than  that  of  A,  since  the 

f 
fraction  — - — ~  is  less  than  unity  ;  but  h  represents  the 

■v/(l+/') 
radius  of  the  cylinder,  and  hence  it  follows  that  the  equi- 


(138). 


140 


STATICS. 


librium  is  only  possible  when  the  distance  r  of  the  point  C 
from  the  direction  of  the  resultant  does  not  exceed  the  radius 
of  the  cylinder.  The  direction  of  the  resultant  will  there- 
fore intersect  the  surface  of  the  cylinder.  This  condition, 
without  which  the  equilibrium  of  the  lever,  maintained  by 
the  effect  of  friction,  becomes  impossible,  is  not  alone  suffi- 
cient ;  for  the  value  of  r  must  not  exceed  that  determined  by 
equation  (140)  ;  otherwise  the  condition  of  moments  could 
not  be  fulfilled. 

278.  It  may  be  remarked,  that  the  equation  of  the  moments 
expresses  the  condition  that  the  friction  and  the  resultant  of 
all  the  forces  applied  to  the  lever,  acting  conjointly,  will  pre- 
vent any  tendency  to  rotation.  For  since  the  direction  of 
the  normal  force  passes  through  the  origin,  it  can  have  no 
tendency  to  produce  rotation.  If,  therefore,  an  equilibrium 
subsists,  it  must  be  produced  in  consequence  of  the  forces 
R  and  F  exerting  equal  efforts  to  turn  the  system  in  con- 
trary directions.  But  this  is  precisely  the  condition  expressed 
by  the  equation  (135),  since  the  moments  of  the  forces  are 
equal  and  have  contrary  signs. 

279.  We  can  also  determine  the  relation  which  must  sub- 
sist between  the  power  and  the  resistance.  For  this  purpose 
the  preceding  results  must  undergo  certain  modifications. 

Let  P  and  S  represent  the  power  and  resistance  [Fig. 
137),  which  form  with  each  other  an  angle  è  ;  the  result- 
ant of  these  two  forces  will  be  determined  by  the  equation 
(Art.  30) 

R2=P2+S2-}-2PScos<i. 

By  substituting  this  value  of  R»  in  equation  (139),  it  becomes 

P2+S2-1-2PS  cosô=N^(l+/2) (141). 

280.  Let  the  value  of  N  be  now  expressed  in  functions  of 
the  quantities  P  and  S.  For  this  purpose,  let  the  perpen- 
diculars f  and  s  be  demitted  on  the  directions  of  the  forces 
P  and  S  respectively  ;  the  moment  Rr  of  the  resultant  can 
then  be  chanored  into  Pp  —  Ss,  or  S5 — Pp,  according  to  the 
direction  in  which  the  resultant  tends  to  turn  the  system  ; 
thus  the  equation  (136)  will  become 

±(Pp-S5)-f-rsyA=o 


FRICTION.  141 

whence, 

and  by  substituting  this  value  in  equation  (141),  we  find 

This  result  may  be  simplified  by  making 

P=S0,     and  — ^=^'' (142). 

The  quantity  S^  will  then  disappear,  being  a  common  fac- 
tor, and  the  equation  will  reduce  to 

z'-{-2z  cos  6-{-l=-^(pz—sy. 

From  this  equation  we  deduce 

z^h-  +2zh'  cos  0+^2  =k^^  {p'z^  —2pzs-\-s^)  ; 
and  by  transposition, 

{k^'p^  —h^)z^  —2{psk^  +/i2  cos  6)z^k^s^—h^=(i\ 
or,  by  division, 

,     21  psk^ -\-h'' cos 6)        k'^s^'—h^      ^ 

z — ~  z  A =U. 

The  value  of  z  deduced  from  this  equation  is  the  ratio  of  the 
power  to  the  resistance  ;  and  since  z  has  two  values,  it  is 
obvious  that  the  first  will  apply  to  the  case  in  which  the 
power  is  about  to  overcome  the  resistance,  and  the  second  to 
that  in  which  the  resistance  is  about  to  overcome  the  power. 
By  resolving  the  equation,  we  find 

_psk'>-]-h'cos6±V[{psk2+h'^cos6y—{k^p^—h'>){k''s^  —  h^)]^ 

and  by  developing  and  reducing  the  terms  contained  under 
the  radical  signs,  we  obtain 

psk^  -\-h^cos6±h^[k^{2r- -\-2pscos6-\-s'')—h''{\—cos''6)]^ 
^'^~~  k'p^-h'  ' 

and  finally,  by  substituting  for  z  and  1  —  cos^  e  their  respect- 
ive values,  we  shall  have 

P     psk'' +h- cos 6 ±hy[k''(p''  +2ps cos 6 +  s'')—h'  sin''  O] 

S  l^2p2_h,2 


142  STATICS. 

281.  If  the  radius  of  the  cyUnder  be  very  small,  its  square 
h^  maybe  neglected,  and  the  preceding  ratio  will  then  become 

P  _5       A^(jJ^+2jJgCOS0  +  5') 

If  the  perpendiculars  p  and  5,  demitted  from  the  point  C  on 
the  respective  directions  of  the  power  and  resistance,  become 
equal  to  each  other,  the  results  will  apply  to  the  case  of  the 
pulley  ;  and  by  still  neglecting  the  quantity  h^,  we  shall  find 

P^       AvW+cos^]  . 

S  kp  ^      ^* 

282.  Finally,  when  the  power  and  resistance  act  in  par- 
allel directions,  the  angle  6  becomes  equal  to  zero  ;  whence, 

sin  ^=0,     cos^=l  ; 
and  the  equation  (143)  then  reduces  to 

?=1±?^. 
S  kp 

283.  The  same  principles  will  serve  to  determine  the  con- 
ditions of  equilibrium  in  the  other  mechanical  powers,  when 
regard  is  had  to  the  effects  of  friction  ;  but  the  results  obtained 
would  in  general  prove  much  more  complicated. 

Of  the  !Stiffness  of  Cordage. 

284.  In  employing  the  cord  as  a  means  of  transmitting  the 
effect  of  a  force  to  a  machine,  we  have  hitherto  supposed  the 
cord  to  be  perfectly  flexible.  But  as  this  hypothesis  is  inad- 
missible in  practice,  it  becomes  necessary  to  estimate  the  ad- 
ditional force  that  will  be  necessary  to  overcome  the  rigidity 
of  the  cord. 

Let  P  and  d  {Fig.  138)  represent  two  weights  which  are 
applied  to  the  extremities  of  a  cord  passing  over  a  fixed  pul- 
ley :  if  the  weight  P  be  supposed  to  prevail,  and  the  cord  be 
regarded  as  perfectly  rigid,  the  extremity  Q.  will  evidently  be 
brought  into  a  position  Q,',  such  that  the  vertical  line  Q'O 
will  intersect  the  horizontal  line  CO  drawn  throusfh  C,  at  a 
distance  CO  from  the  centre,  greater  than  the  radius  CG. 
The  extremity  P  will  at  the  same  time  assume  the  position  P', 


STIFFNESS    OF   CORDAGE.  143 

such  that  the  vertical  Hne  drawn  through  P'  will  intersect 
the  radius  CF.  Hence  the  arm  of  the  lever  to  which  the 
force  Q,  is  applied  will  now  be  longer  than  that  of  the  force 
P,  and  the  condition  of  equilibrium  will  therefore  require  that 
the  force  P  shall  exceed  Q., 

285.  If  the  cord  be  supposed  imperfectly  rigid,  similar 
effects  will  be  produced,  though  in  a  less  degree  ;  and  in 
practice,  it  is  found  that  the  decrease  in  the  arm  of  lever,  to 
which  the  preponderating  weight  is  applied,  is  wholly  insen- 
sible. Hence,  in  estimating  the  effects  produced  by  the  rigid- 
ity of  a  cord  employed  in  a  machine,  it  will  simply  be  neces- 
sary to  increase  the  arm  of  the  lever  to  which  the  resistance 
Q  is  applied,  by  a  proper  quantity  q. 

286.  To  determine  the  value  of  q^  we  remark  that  the  re- 
sistance to  flexure  opposed  by  a  given  cord  arises  from  two 
distinct  causes, — viz.  1°.  The  tension  of  the  cord,  or  the  force 
Q,  which  is  employed  to  stretch  it  ;  and,  2°.  The  materials 
used  in  the  construction  of  the  cord,  and  the  degree  of  twist 
which  has  been  given  to  it.  The  resistance  arising  from  the 
tension  of  the  cord  is  found  to  be  proportional  to  this  tension, 
and  may  therefore  be  represented  by  an  expression  of  the 
form  6Q,  in  which  h  represents  an  indeterminate  constant. 
The  resistance  produced  by  the  second  cause  may  be  repre- 
sented by  a  quantity  a. 

Thus,  for  the  same  cord  bent  over  the  same  pulley,  the 
expression  («  +  6Q,)  may  be  supposed  to  represent  the  effort 
necessary  to  bend  it.  But  if  we  suppose  the  diameter  of  a 
second  cord  to  be  greater,  the  force  necessary  to  bend  it  will 
become  greater,  and  we  can  assume  that  this  force  will 
increase  according  to  some  power  n  of  the  diameter  D.  The 
force  will  also  increase  as  the  curvature  increases,  or  as  the 

radius  of  the  pulley  is  decreased,  and  hence (a +  60,)  may 

be  taken  as  an  expression  for  the  force  necessary  to  overcome 
the  rigidity  of  the  cord.  This  expression  represents  the 
increment  that  must  be  given  to  the  power  P,  in  order  that 
it  may  be  on  the  point  of  overcoming  the  resistance  Q,  :  but 
we  also  have 

Vr=Gi{r-\-q)  ; 


144  STATICS. 

and  since  the  forces  P  and  Q,  become  equal  when  the  cord  is 

supposed  destitute  of  rigidity,  P— Q,  or  Q  i  will  also  express 

r 

the  value  of  this  increment.     By  making  these  values  equal 

to  each  other,  we  obtain 

D"(a+6a)=%; 

whence, 

q^^l.{a-\-hGi) (143  a). 

287.  This  equation  should  only  be  regarded  as  furnishing 
an  approximate  value  of  the  quantity  q,  since  the  above  rela- 
tion has  been  obtained  by  considerations  of  a  very  general 
character.  It  moreover  contains  certain  unknown  quantities 
a,  b,  and  ?i,  which  vary  with  different  cords. 

For  the  purpose  of  verifying  the  truth  of  the  preceding 
formula,  and  at  the  same  time  determining  the  values  of  the 
unknown  constants,  we  proceed  as  follows. 

Having  selected  a  cord,  we  pass  it  over  a  fixed  pulley,  and 
attach  to  its  extremities  two  equal  weights  :  we  then  increase 
one  of  these  weights  until  it  is  about  to  prevail  over  the  other, 
and  the   difference  k  will  give  one  value  of  the  quantity 

r 

By  repeating  the  experiment  several  times,  changing  the 
weights,  the  cord,  or  the  pulley,  we  can  obtain  a  number  of 
similar  equations,  in  which  the  quantities  a,  b,  and  n  will  be 
the  same,  and  the  quantities  D,  r,  and  Q.,  although  different, 
will  be  known  by  observation.  Three  such  equations  will 
serve  to  determine  a,  b,  and  7i,  and  their  values  being  sub- 
stituted in  the  general  relation  expressed  by  formula  (143  a), 
the  accuracy  of  the  formula  can  be  tested  by  comparing  it 
with  the  results  furnished  by  other  experiments. 

The  quantity  n  was  found  by  Coulomb  to  be  usually  about 
1.7  or  1.8  ;  and  the  resistance  to  flexure  must  therefore  vary 
nearly  as  the  square  of  the  diameter  of  the  cord  :  but  the 
quantity  n  is  itself  subject  to  some  variation,  becoming  nearly 

1.4  when  the  cord  has  been  long  used. 

The  following  results,  expressed  in  French  poinids,  were 

obtained  in  the  experiments  of  Coulomb. 


RESISTANCE    OP   SOLIDS.  145 


ibs.  Tin  lbs. 


J)rt  iOS.  jy„ 

30  threads  in  a  yarn  .  .  —  Xa=4.2 


White  rope     <  15  threads 
f    6  threads 


Tarred  rope 


30  threads  in  a  yarn  .  .   —  Xa=6.6 


15  threads 
6  threads 


i4.2  . 

.     _ixl00=:  9 
r 

1.2  . 

.  .  .  .  " 5.1 

0.2  . 

« 2.2 

lbs. 
;6.6   . 

D»                 lbs. 
.   —6X100=11.6 
r 

2.0  . 

« 5.6 

0.4  . 

" 2.4 

On  the  Resistance  of  /Solids. 

2S8.  The  particles  of  every  solid  body  are  found  to  oppose 
a  certain  resistance  to  any  force  which  tends  to  separate  them. 
This  resistance  arises  from  the  mutual  actions  exerted  by  the 
particles  upon  each  other  ;  and  if  the  nature  of  these  actions, 
as  well  as  the  arrangement  of  the  particles  which  compose 
the  body,  were  accurately  known,  it  might  be  possible  to 
estimate  the  force  necessary  to  separate  the  particles,  or  to 
produce  a  given  change  in  the  figure  of  the  body.  Bat  as  we 
are  entirely  ignorant  of  these  particulars,  it  becomes  neces- 
sary to  adopt  some  hypothesis  relative  to  the  manner  in  which 
bodies  are  constituted,  and  the  nature  of  the  actions  exerted 
by  the  particles  upon  each  other.  Then,  by  reasoning  upon 
such  hypothesis,  we  can  obtain  results  which,  compared  with 
those  derived  from  experiment,  will  serve  to  test  the  accuracy 
of  the  supposition. 

289.  The  hypotheses  most  generally  adopted  are — 1'^,  That 
of  Galileo,  which  supposes  all  solid  bodies  to  be  niade  up  of 
fibres,  disposed  parallel  to  the  length  of  the  body,  and  sus- 
ceptible of  being  ruptured  without  undergoing  flexure,  ex- 
tension, or  compression  ;  or,  2°.  That  of  Leibnitz,  modified 
by  Bernoulli  and  others,  which  regards  the  fibres  of  all  bodies 
as  elastic  ;  being  susceptible  of  extension  and  compression,  and 
capable  of  opposing  a  resistance  directly  proportional  to  their 
extensions  or  compressions.  The  force  required  to  produce 
a  given  extension  is,  moreover,  supposed  to  be  equal  to  that 
which  is  capable  of  producing  an  equal  compression. 

290.  It  is  very  certain  that  neither  of  these  hypotheses  is 
strictly  correct  ;  but  as  the  results  given  by  the  latter  difier  but 

K  13 


146  STATICS. 

little  from  the  truth,  when  the  extensions  or  compressions  are 
inconsiderable,  we  shall  adopt  it,  and  apply  it  to  the  investiga- 
tion of  the  resistance  which  a  solid  will  oppose  under  different 
circumstances. 

291.  The  kind  of  resistance  which  the  body  offers  will  de- 
pend in  a  great  measure  upon  the  manner  in  which  the  force  is 
applied.  Thus,  the  force  may  exert  an  effort  to  extend  or 
compress  the  solid  in  the  direction  of  its  length,  or  it  may 
tend  to  produce  a  flexure  of  the  solid,  or  it  may  operate  as 
a  force  of  torsion  ;  and  in  each  of  these  cases  it  may  be 
required  to  determine  the  force  necessary  to  produce  a  rupture 
or  separation  of  the  particles,  or  simply  that  necessary  to 
effect  a  given  change  in  the  figure  of  the  solid. 

The  cases  which  more  generally  occur  are,  1°.  That  in 
which  the  solid  sustains  an  extension  or  compression  in  the 
direction  of  its  length,  without  undergoing  sensible  flexure  ; 
and,  2°.  That  in  which  flexure  is  produced  by  the  applica- 
tion of  a  force  perpendicular  to  the  length  of  the  solid. 

As  it  is  the  object  of  the  present  article  merely  to  exhibit 
the  general  methods  in  which  the  hypothesis  assumed  may 
be  applied  to  the  determination  of  the  strength  of  bodies,  or 
the  resistance  which  they  are  capable  of  opposing,  we  shall 
confine  our  investigations  to  the  consideration  of  these  two 
cases. 

292.  The  resistance  of  a  body  to  a  change  of  figure  de- 
pends upon  its  force  of  elasticity,  which  is  measured  by  the 
effort  necessary  to  compress  or  extend  the  body  by  a  given 
quantity.  Its  resistance  to  rupture  depends  upon  its  force  of 
tenacity,  or  upon  the  effort  necessary  to  rupture  or  crush  the 
body. 

The  values  of  these  forces  having  been  determined  experi- 
mentally for  a  body  composed  of  a  given  substance,  and 
having  a  simple  form,  we  can  calculate  the  compression,  ex- 
tension, or  flexure  produced  in  another  body,  of  the  same 
substance,  by  the  application  of  a  given  force.  The  methods 
of  effecting  this  calculation  will  now  be  explained. 


RESISTANCE   OF   SOLIDS.  147 

Of  the  Resistance  to  Compression  or  Extension. 

293.  When  a  solid  is  stretched  or  compressed  in  the  direc- 
tion of  its  length,  being  at  the  same  time  prevented  from 
experiencing  flexure,  the  lengths  of  its  fibres  are  found  to 
undergo  very  slight  variations,  and  we  can  therefore  assume, 
in  conformity  with  the  hypothesis  adopted,  1°.  That  the 
extensions  or  compressions  of  all  the  fibres  will  be  equal  to 
each  other,  and  uniform  throughout  the  extent  of  each  fibre  ; 
and  that  the  force  necessary  to  produce  a  given  extension  will 
be  capable  of  producing  an  equal  compression.  2°.  That 
the  variations  in  the  lengths,  and  the  resistances  opposed  by 
the  fibres,  are  constantly  proportional  to  the  forces  which 
produce  them  ;  and  that  this  proportion  obtains  even  for  those 
forces  which  rupture  or  crush  the  body. 

294.  Let  a  cubical  mass  of  any  substance  be  placed  upon 
a  horizontal  plane,  and  subjected  to  the  action  of  a  weight 
which  rests  upon  its  upper  surface,  compressing  the  substance 
in  the  vertical  direction.     Denote  by 

a,  the  length  of  one  of  the  edges  of  the  cube  ; 

a',  the  quantity  by  which  its  vertical  dimension  is  com- 
pressed, and  which  is  always  extremely  small  in 
comparison  with  a  ; 

P,  the  force  which  produces  the  compression. 
Then,  since  the  compression  of  each  fibre  is  supposed  uni- 
form throughout,  or  since  the  particles  which  compose  any 
one  fibre  are  supposed  to  approach  each  other  equally  at  every 
point  of  such  fibre  ;  it  is  obvious  that  the  entire  compression 
a',  sustained  by  any  fibre,  will  be  directly  proportional  to  its 
length  a.  For  example,  if  the  length  of  another  solid  be 
supposed  equal  to  2a,  its  transverse  section  remaining  the 
same,  and  if  the  same  force  P  be  applied  to  its  upper  surface, 
the  number  of  particles  in  the  length  2a  will  be  twice  as  great 
as  the  number  contained  in  a  ;  and  each  pair  of  consecutive 
particles  being  caused  to  approach  each  other  to  within  the 
same  distance,  in  order  that  the  resistance  of  the  fibre  may 
be  uniform  throughout,  the  whole  variation  in  the  length  2a 
will  evidently  be  twice  as  great  as  that  which  was  produced 

K2 


148  STATICS. 

in  the  length  a,  and  will  therefore  be  expressed  by  2a'.  And, 
generally,  the  compression  of  the  solid,  whose  length  is  7ia, 
and  whose  transverse  section  remains  the  same,  will  be  ex- 
pressed by  im',  when  the  same  force  P  is  applied  to  its  upper 
surface.  Let  the  quantity  a  be  supposed  equal  to  the  linear 
unit, — one  foot,  for  example  ;  then  71  will  express  the  number 
of  feet  contained  in  the  length  of  the  second  solid,  and  7ia' 
will  express  the  variation  produced  in  the  length  of  a  solid 
whose  transverse  section  is  equal  to  one  square  foot,  and 
whose  length  is  equal  to  71  feet. 

295.  The  preceding  remarks  have  been  confined  to  the 
case  in  which  the  solid  suffers  compression,  but  from  the 
nature  of  the  hypothesis, they  must  apply  with  equal  force  to 
the  case  in  which  the  eflbrt  is  exerted  to  extend  the  body. 

296.  If  the  transverse  section  of  a  second  solid,  whose 
length  is  likewise  equal  to  7i,  be  supposed  greater  than  that 
of  the  first,  the  number  of  its  fibres  will  be  increased  in  the 
same  proportion,  and  the  total  effort  exerted  by  these  fibres 
when  compressed  to  the  same  degree  will  evidently  be  pro- 
portional to  their  number  :  thus,  if  P'  represent  the  force 
necessary  to  compress  a  prism  whose  length  is  71,  and  whose 
transverse  section  contains  ??i  square  feet,  by  a  quantity  equal 
to  na,  we  shall  have  the  proportion 

section  1  :  section  w^  :  :  P  :  P'  ; 
whence, 

P'=wP. 

297.  If  the  force  P'  be  increased,  the  solid  will  undergo  a 
greater  compression,  and  the  quantity  by  which  the  length  /* 
of  the  fibre  is  compressed  will  no  longer  be  represented  by 
'iia\  but  by  an  unknown  quantity  7ia".  To  determine  this 
quantity,  we  recur  to  the  hypothesis  which  assumes  that  the 
compressions  are  proportional  to  the  forces  which  produce 
them  ;  hence,  by  calling  P"  the  value  of  the  force  which  pro- 
duces the  compression  71a",  we  shall  have 

7ia'  :  7ta"  :  :  P'  :  P", 
and  therefore, 

P"=P'Z^- 

na'  ' 


RESISTANCE   OP   SOLIDS.  149 

or,  replacing  P'  by  its  value  mP,  we  have 

p„^P^m^" ^^3jj 

d         n 
p 

298.  The  quantity  —  is  called  the  coefficient  of  the  elas- 

a 

ticity  :  its  value  will  depend  only  on  the  elastic  force  of  the 
substance  of  which  the  prism  is  composed,  and  will  therefore 
be  independent  of  the  dimensions  of  the  particular  prism 
under  consideration.  If  we  denote  this  coefficient  by  A,  we 
shall  obtain,  for  the  entire  compression  of  the  prism, 

„     nV" 

na  = — r-. 

mA 

This  expression  will  determine  the  quantity  by  which  a  given 
prism  will  be  compressed  under  the  influence  of  a  given  force, 
when  the  coefficient  of  the  elasticity  has  been  previously 
ascertained.  It  should  be  remembered,  however,  that  this 
formula  is  only  applicable  when  the  compressions  are  exceed- 
ingly small  ;  and  that  the  solid  is  ruptured  or  crushed  before 
its  length  undergoes  a  very  sensible  change. 

299.  The  preceding  expression  is  equally  applicable  when 
the  force  P"  tends  to  stretch  the  solid. 

300.  To  determine  the  force  necessary  to  rupture  a  given 
prism,  when  exerted  in  the  direction  of  the  length  of  the 
prism,  we  shall  denote  by  B  the  force  necessary  to  rupture 
a  prism  of  the  given  substance  whose  transverse  section  is  a 
square  foot.  Then,  if  the  transverse  section  of  the  given 
prism  be  supposed  to  contain  m  square  feet,  the  number  of 
its  fibres  will  be  m  times  greater  than  the  number  contained 
in  the  prism  whose  section  is  equal  to  one  square  foot  ;  and 
since  each  fibre  in  the  two  prisms  must  oppose  the  same 
resistance  at  the  instant  of  rupture,  we  shall  determine  the 
force  P"  necessary  to  rupture  the  given  prism,  by  the  propor- 
tion 

section  1  :  section  w  :  :  B  :  P"  ; 
whence, 

P"=mB. 

301.  The  quantity  Bis  called  the  coefflciejit  of  the  tenacity, 
and  depends  only  on  the  nature  of  the  substance  under  consid- 


150 


STATICS. 


eration.  Having  determined  ihis  quantity  by  experiment,  we 
can  readily  calculate  the  force  necessary  to  rupture  a  given 
prism  of  the  same  substance.  This  investigation  is  equally 
applicable  whether  the  force  be  exerted  to  compress  or  extend 
the  solid.  The  methods  of  determining  experimentally  the 
coefficients  of  the  elasticity  and  tei.acity  will  be  explained 
hereafter. 

Of  the  Resistance  of  a  Solid  to  Flexure  and  Fracture  produced 
by  a  Force  acting  at  right  angles  to  the  direction  of  the 
Fibres. 

302.  When  the  length  of  a  solid  body  bears  a  certain  pro- 
portion to  its  thickness,  the  body  is  found  to  undergo  a  cer- 
tain degree  of  flexure  before  breaking.  Th^s  flexure  becomes 
more  perceptible  as  the  length  of  the  solid  is  increased  :  thus 
a  bar  of  wrought  iron  whose  length  does  not  exceed  twelve 
or  fifteen  times  its  thickness  gives  very  slight  indications  of 
flexibility  ;  but  when  its  length  is  increased  to  forty  or  fifty 
times  its  thickness,  it  yields  readily  to  an  effort  exerted  to 
bend  it,  and  becomes  susceptible  of  taking  a  very  consider- 
able flexure  before  breakingf. 

303.  If  a  force  P  be  applied  in  a  direction  perpendicular 
to  the  length  of  the  solid  AB  {Fig.  139),  which  is  supported 
at  its  two  extremities,  and  if  this  force  be  supposed  to  produce 
a  certain  degree  of  flexure  in  the  solid,  causing  it  to  assume 
the  form  represented  in  Fig.  139  a,  the  fibres  aa,  &c.  situated 
on  the  convex  side  will  be  extended,  their  lengths  being  in- 
creased, and  those  situated  on  the  concave  side  will  suffer  a 
compression,  and  will  undergo  a  diminution  in  length.  This 
effect  is  readily  observed  :  for,  if  the  force  P  be  gradually 
increased  until  it  become  capable  of  breaking  the  solid,  the 
rupture  will  be  found  to  commence  at  a  point  D  on  the  con- 
vex side,  thereby  indicating  that  the  fibres  aa  on  that  side 
have  been  most  extended  ;  and  if  some  of  the  fibres  situated 
on  the  convex  side  be  previously  separated  by  cutting  them 
through  transversely,  it  will  be  found  that  a  smaller  force 
than  P  will  be  required  to  fracture  the  solid.  But  if,  on  the 
contrary,  the  fibres  bb  situated  near  the  opposite  side  of  the 


RESISTANCE    OP   SOLIDS.  161 

solid  be  cut  transversely  to  a  certain  depth  EF  (Fig.  139), 
and  if  a  thin  plate  of  some  unyielding  substance  be  intro- 
duced into  the  cut  EF,  so  as  to  fill  it  entirely,  it  will  be  found, 
upon  subjecting  the  solid  to  the  action  of  the  force  P,  that 
the  thin  plate  will  be  -retained  by  a  strong  pressure  tending  to 
compress  it,  and  that  the  strength  of  the  solid  will  not  be 
diminished,  the  rupture  commencing  at  the  convex  side, 
when  the  force  P  has  been  increased  in  the  same  degree  as 
was  necessary  to  rupture  the  solid  before  severing  any  of  its 
fibres. 

As  we  proceed  from  the  convex  towards  the  concave  side 
of  the  solid,  the  extensions  of  the  fibres  will  gradually  dimin- 
ish, and  at  a  certain  distance  from  the  surface,  their  lengths 
will  undergo  no  variation  ;  beyond  this  distance  the  exten- 
sions will  be  changed  into  compressions,  and  these  will  again 
increase  until  we  arrive  at  the  concave  side. 

304.  The  flexure  of  the  fibres  being  supposed  to  take  place 
entirely  in  planes  parallel  to  the  axis  of  the  solid  and  the 
direction  of  the  force  applied,  it  is  evident  that  the  change  of 
figure  experienced  by  the  solid  will  require  that  those  fibres 
whose  lengths  undergo  no  variation  should  be  contained, 
previous  to  the  flexure,  in  a  plane  perpendicular  to  the  direc- 
tion of  the  force  which  produces  the  flexure  ;  and  that,  after 
the  flexure,  these  fibres  will  form  a  cylindrical  surface, 
whose  elements  will  be  parallel  to  the  same  plane.  More- 
over, the  fibres  situated  at  equal  distances  from  this  plane 
will  undergo  equal  extensions  or  compressions. 

305.  Let  us  now  conceive  a  right  prism  AB  to  be  firmly 
fixed  at  its  extremity  A,  in  such  manner  that  its  axis  shall  be 
horizontal,  and  that  a  vertical  plane  passing  through  the  axis 
shall  divide  the  solid  into  two  symmetrical  parts.  Let  a 
weight  P  be  applied  at  the  other  extremity  of  the  solid, 
causing  it  to  undergo  a  certain  degree  of  flexure,  and  to 
assume  the  form  represented  in  Fig.  140.  If  two  planes, 
ariv,  a'u'v',  be  drawn  infinitely  near  to  each  other,  and  normal 
to  the  curve  Auu'B  assumed  by  the  fibres  whose  lengths  re- 
main invariable,  such  planes  will  include  between  them  an 
elementary  portion  of  the  solid,  and  if  the  system  be  sup- 
posed in  equilibrio,  the  state  of  equilibrium  will  not  be  dis- 


162  STATICS. 

turbed  by  regarding  the  portion  of  the  solid  included  between 
the  sections  ACD  and  uav  as  absolutely  immoveable,  and 
the  portion  of  the  solid  included  between  the  sections  BEF 
and  2iav  as  constituting  a  distinct  system.  The  conditions 
of  equilibrium  in  this  system  will  evidently  require  that  the 
force  Pj  together  with  the  force  necessary  to  retain  the  part 
DCAïiav  in  its  position,  shall  be  just  capable  of  sustaining 
the  efforts  arising  from  the  compressions  and  extensions  of 
the  fibres,  or,  in  other  words,  that  all  these  forces  should  reduce 
to  two  that  are  equal  to  each  other  and  directly  opposite. 

306.  If  we  assume  any  two  rectangular  axes  Ax  and  Ay 
situated  in  the  vertical  plane  passing  through  the  axis  of  the 
solid,  we  can  resolve  each  of  the  several  forces  into  two  com- 
ponents respectively  parallel  to  these  axes  ;  since  these  forces 
are  all  situated  in  planes  parallel  to  the  plane  of  the  axes. 
Moreover,  since  the  solid  has  been  supposed  to  be  symmetri- 
cally divided  by  the  vertical  plane  passing  through  the  axis, 
the  forces  of  elasticity  arising  from  the  extensions  or  com- 
pressions of  the  different  fibres  will  be  symmetrically  disposed 
with  respect  to  this  plane,  and  the  conditions  of  equilibrium 
will  therefore  be  the  same  as  though  the  forces  were  all  situ- 
ated in  this  plane.  These  conditions  are,  1°.  That  the  sum 
of  the  components  parallel  to  each  axis  shall  be  equal  to  zero  ; 
and,  2°.  That  the  sum  of  the  moments  of  all  the  forces  taken 
with  respect  to  any  line  perpendicular  to  the  plane  of  the 
forces  shall  be  equal  to  zero. 

307.  We  shall  assume  the  origin  of  co-ordinates  at  the  fixed 
extremity  A  of  the  solid,  and  refer  the  points  in  the  curve 
Ann'  Bto  the  axes  of  x  and  y,  which  are  respectively  hori- 
zontal and  vertical. 

308.  The  normal  plane  auv  intersects  the  cylindrical  sur- 
face which  contains  the  fibres  of  an  invariable  length,  and  the 
vertical  plane  passing  through  the  axis  of  the  solid,  in  two 
lines  au  and  nv,  at  right  angles  to  each  other  ;  and  the  points 
in  the  section  anv  will  be  referred  to  two  rectangular  axes, 
one  of  which  au  will  be  called  the  axis  of  ii,  and  the  other, 
parallel  to  irv,  and  passing  through  the  origin  a,  will  be  desig- 
nated as  the  axis  of  v.  Thus  the  two  co-ordinates  of  the 
point  7)1  will  be  ao=u,  and  om=v.     The  moments  of  the  sev- 


RESISTANCE   OF   SOLIDS.  153 

eral  forces  will  be  referred  to  the  line  au,  which  is  frequently 
called  the  axis  of  equilibrium. 

309.  This  being  premised,  we  shall  denote  by 

A  and  B,  the  coefficients  of  elasticity  and  tenacity  (Arts. 

298  and  301), 
R,  the  radius  of  curvature  wr,  of  the  curve  of  flexure,  at 

the  point  u, 
s,  the  length  of  the  arc  Au  of  the  curve  of  flexure, 
X  and  y,  the  co-ordinates  Ap  and  jm  of  the  point  u  re- 

ferred  to  the  origin  A, 
x'  and  y',  the  co-ordinates  of  the  point  ^  referred  to  the 

same  origin, 
U  and  U',  functions  of  the  absciss  ao=t(.,  expressing  the 
values  of  the  corresponding  ordinates  ol  and  ol'  of 
the  curve  of  intersection,  reckoned  from  the  axis 
of  equilibrium  au,  towards  the  convex  and  concave 
sides  of  the  solid, 
a,  the  dimension  of  the  solid  estimated  along  the  axis 

of  equilibrium, 
V,  the  greatest  value  of  U  or  U',  or  the  distance  from  the 
axis  of  equilibrium  to  that   fibre  which   is  most 
stretched  or  compressed  at  the  instant  of  rupture. 
Then,  if  we  consider  an  elementary  portion  of  the  solid,  in- 
cluded between  the  consecutive  normal  planes,  whose  base  is 
represented  by  the  element  inm" = du.  dv,  of  the  normal  section 
auv,  its  original  length  will  be  equal  to  uu'=ds  ;  and  after 
the  flexure,  this   length  will  be  increased    or   diminished, 
according  to  its  position  with  reference  to  the  axis  of  equili- 
brium, and  will  be  represented  by  mm'  or  nn'  {Fig.  141). 
But  from  similarity  of  the  figures  rmm\  rim',  run',  we  have 
the  proportion 

rii  :  rm  :  m  :  :  uu'  :  mm'  :  nn'  \ 
or, 

R  :  R  +  v  :  R — v  :  :  uu'  :  inmf  :  w/i'  ; 
and  therefore, 

R  :  V  :  r  :  :  uu'  :  mm' — uu'  :  uu!  —  nn'  \ 
whence, 

,    v.uu'    vds 
mm  —uiv=tiu—nn=  -5—  =  -5-. 
R         R 


154  STATICS. 

This  expression  will  represent  the  variation  in  the  length  of 
the  element  whose  base  is  equal  to  du.dv  {Pig.  140)  and 
whose  original  length  was  equal  to  ds.  To  determine  the 
resistance  opposed  by  this  element  when  thus  extended  or 
compressed,  we  employ  the  expression  (143  b)  in  which  we 
replace  7ia",  the  variation  in  length,  by  mm'—uu',  or  nu'—nn'  ; 
the  transverse  section  ni,  by  dii .  dv  ;  and  the  length  ii,  by 
ds  :  we  shall  thus  obtain  an  expression  for  the  resistance  P" 
opposed  by  the  element, 

jy„_vds     dudvK _Avdvdu  ,-...y   v 

"^  ~R^    ds KT" ^       ''^'' 

and  the  moment  of  this  resistance  taken  with  reference  to  the 
axis  of  equilibrium  au,  will  be 

A  A 

-vdvdu'><.v  =  -v^dvdu (143  d). 

SX  K 

310.  The  other  elementary  portions  of  the  solid  included 
between  the  consecutive  normal  planes  will  give  similar 
expressions  for  the  resistances  and  their  moments  ;  and  by 
taking  the  sums  of  these  expressions,  we  shall  obtain  the 
value  of  the  entire  resistance,  and  that  of  its  moment  with 
reference  to  the  axis  au.  To  determine  the  value  of  these 
sums,  we  must  integrate  the  expressions  (143  c)  and  (143  d) 
throuo^hout  the  limits  of  the  section  a7iv.  This  intégration 
is  effected,  first  with  reference  to  one  of  the  variables,  v  for 
example  ;  and  its  value  being  then  substituted  in  terms  of  u, 
we  integrate  a  second  time  with  reference  to  the  other  varia- 
ble.  The  limits  of  the  first  integration  will  evidently  be 
v=0,  and  ^=11,  for  those  fibres  which  sufier  extension  ;  and 
v=Q,  v—V,  for  those  which  suffer  compression.  The  limits 
of  the  second  integration  will  be  7i=0,  and  u=a. 

311.  This  being  premised,  the  sum  of  the  resistances 
arising  from  the  extensions  of  the  several  fibres  will  be  ex- 
pressed by 

*  An  expression  of  the  form  y  du  is  intended  to  indicate  that  the  integral  of 
da  is  to  be  taken  between  the  limits  u=0,  and  u=a.  In  like  manner,  f  vdv  sig- 
nifies that  the  integral  of  vdv  shonld  be  taken  between  the  limits  v=0,  and  v=U. 


RESISTANCE    OP   SOLIDS.  155 

and  the  sumofthe  resistances  arising  from thecompressions  of 
the  fibres  will  be 

The  sum  of  the  moments  of  these  resistances,  taken  with 
reference  to  the  axis  au,  will  be 

£(/"''"/''"■*+/"''"/"'"'''') (1*3^)- 

312.  For  the  purpose  of  resolving  the  resistances  (143  e) 
and  (143/)  into  components  parallel  to  the  axes  of  x  and  y, 

we  must  multiply  them  respectively  by  -—  and--^,  the  cosines 

CiS  (tS 

of  the  angles  which  their  directions  form  with  the  axes  :  but 
as  the  curvature  assumed  by  the  solid  is  always  found  to  be 
exceedingly  small  even  at  the  instant  when  the  rupture  takes 

place,  the  expression  -—  will  be  very  nearly  equal  to  unity, 
as 

and  the  components  in  the  direction  of  the  axis  of  x  may  there- 
fore be  assumed  equal  to  the  entire  resistances.  These  being 
the  only  forces  in  the  system  which  have  components  par- 
allel to  the  axis  of  x,  the  condition  of  equilibrium  which 
requires  that  the  sum  of  the  components  parallel  to  this  axis 
shall  be  equal  to  zero,  will  be  expressed  by  the  equation 

/a         /»U                  pa          /»U' 
duj     vdv—J    duj    vdv^O (143 /t). 

The  negative  sign  is  given  to  the  resistances  offered  by  those 
fibres  which  suffer  compression,  because  they  are  exerted  in  a 
direction  contrary  to  the  resistances  of  the  extended  fibres. 

This  equation  will  determine  the  position  of  the  axis  of 
equilibrium  au  when  the  figure  of  the  transverse  section  is 
known. 

313.  A  similar  condition  may  be  obtained  for  the  com- 
ponents parallel  to  the  axis  of  y]  but  as  it  will  not  be  required 
in  the  succeeding  steps  of  this  investigation,  it  will  be  unne- 
cessary to  express  it  analytically. 

314.  The  moment  of  the  force  P  taken  with  reference  to 
the  axis  au  will  be  expressed  by  V{x'—x),  and  since  this 
force  tends  to  turn  the  system  about  the  axis  au,  in  a  direô- 


156 


STATICS. 


tion  contrary  to  that  in  which  the  resistances  of  the  fibres 
would  cause  it  to  turn,  the  condition  that  tfie  algebraic  sum 
of  the  moments  of  all  the  forces  taken  with  reference  to  the 
axis  of  equilibrium  shall  be  equal  to  zero,  will  be  expressed 
by  the  equation 

^{jduj    v^dv^jdaj     v^'dvS —V{x' —x)^{)  .{\^Zi). 

315.  When  the  radius  of  curvature  becomes  equal  to  unity 
the  expression  (143  g)  becomes 

KyJ    duj    v^dv^-J    dvj     v-dv) (143 A-). 

This  quantity  is  called  the  moment  of  elasticity  of  the  solid, 
and  will  depend  upon  the  elasticity  of  the  substance,  and  the 
figure  of  the  transverse  section.  Its  value  will  evidently 
determine  that  of  the  force  P,  which,  acting  at  the  extremity 
of  a  given  arm  of  lever,  will  be  necessary  to  produce  a 
given  curvature  in  the  solid  ;  thus,  the  moment  of  elasticity 
becomes  a  proper  measure  of  the  resistance  to  flexure  opposed 
by  the  solid. 

316.  If  the  flexure  of  the  solid  be  supposed  such  that  the 
extreme  fibre,  or  that  which  undergoes  the  greatest  extension 
or  compression,  is  about  to  be  ruptured  or  crushed,  the  resist- 
ance opposed  by  this  fibre  will  be  that  due  to  the  tenacity  of 
the  substance  :  hence,  if  dudv  denote,  as  in  Art.  309,  the  base 
of  an  elementary  portion  of  the  solid  included  between  the 
consecutive  normal  sections,  and  if  the  distance  of  this  ele- 
ment of  the  solid  from  axis  of  equilibrium  au  be  equal  to  V, 
that  of  the  fibre  which  is  most  extended  or  compressed,  the 
resistance  opposed  by  such  element  will  be  expressed  by 

Bdiidv  ; 
B  denoting  the  coefficient  of  the  tenacity. 

This  element  being  at  the  distance  V  from  the  axis  of 
equilibrium,  its  original  length  ds  will  undergo  a  variation 
represented  (Art.  309)  by 

Yds 
R  ' 
tind  the  corresponding  variation  in  the  length  ds  of  the  ele- 


RESISTANCE    OF    SOLIDS.  157 

ment,  whose  distance  from  the  same  axis  is  denoted  by  v, 
will  be 

vds 

~K' 
But  the  resistances  opposed  by  the  two  elements  being  by 
hypothesis  (Art.  293),  proportional  to  their  extensions  or  com- 
pressions, we  shall  have  the  proportion 

Yds     vds        r»j    J        r»// 
—5-  :  -5-  :  :  Bdudv  :  P", 
SX         JK 

P"  denoting  the  resistance  opposed  by  the  element  at  the 
distance  v  from  the  axis  of  equilibrium.  From  this  propor- 
tion we  deduce 

T?"=l.Bdudv. 

317.  Similar  expressions  may  be  obtained  for  the  resistances 
offered  by  the  other  elements  ;  and  by  taking  their  moments 
with  reference  to  the  axis  of  equilibrium,  and  adding  them 
mto  one  sum,  we  shall  obtain  for  the  moment  of  the  entire 
resistance,  at  the  instant  when  a  fracture  commences. 

This  expression  is  called  the  moment  of  rupture,  and  will 
depend  upon  the  tenacity  of  the  substance,  and  the  figure  of 
the  transverse  section.  This  moment  must  evidently  be 
equal  to  the  moment  V[x' — x)  of  the  force  P,  which  is  just 
capable  of  causing  rupture.     Thus,  we  shall  have 

Y{/,"di/\^^dv+fJduf\^dv'^=V{x'-x)...{U3m). 

The  value  of  the  moment  of  rupture  will  serve  to  determine 
that  of  the  force  P,  which,  acting  at  the  extremity  of  a  given 
arm  of  lever,  will  be  just  capable  of  producing  fracture. 
Thus,  the  moment  of  rupture  becomes  a  proper  measure  of 
the  resistance  to  fracture  opposed  by  the  solid. 

318.  By  comparing  the  expression  (143  k),  for  the  moment 
of  elasticity,  with  (143  I),  which  represents  the  moment  of 
rupture,  we  shall  perceive  that  the  latter  may  be  deduced  from 

the  former  by  merely  substituting  —  for  A. 

41  ^ 


158  STATICS. 

319.  When  the  transverse  section  can  be  divided  sym- 
metrically by  a  horizontal  line,  that  line  will  be  the  axis  of 
equilibrium,  since  the  equation  (143  A)  will  evidently  be  satis- 
fied by  regarding  that  line  as  the  axis  of  ti.  The  moment 
of  elasticity  will  then  be  expressed  by 


2Af    dut    v-clv] 

t/  0        t/  0 


and  the  moment  of  rupture  by 

— -  /     dut     v^dv, 

320.  In  other  cases,  it  will  be  necessary  to  determine  the 
position  of  the  axis  of  equilibrium  by  the  condition  (143 /i), 
and  then  to  calculate  separately  the  two  integrals  which  enter 
into  the  expressions  for  the  moments  of  elasticity  and  rup- 
ture. 

321.  To  apply  these  principles,  we  shall  determine  the 
moments  of  elasticity  and  rupture  for  those  solids  whose 
transverse  sections  are  such  as  are  more  commonly  adopted 
in  practice. 

322.  Let  the  transverse  section  be  a  rectangle  {Fig.  142), 
whose  breadth  and  height  are  denoted  respectively  by  a  and  b. 
The  value  of  the  moment  of  elasticity  will  then  become 

2  At     dut      V'dv. 


pa         pib 

At     dut     V' 

•/  0        t/  0 


and  by  integrating  with  reference  to  v=om,  between  the  limits 
v=0,  and  v=ot=^b,  we  shall  obtain  double  the  sum  of  the 
moments  of  all  the  elements,  whose  bases  constitute  the  ele- 
mentary rectangle  oq.     Performing  the  integration,  we  have 

2Ar  duy."^ =2Ar  duy.^^l=i-^Ab^r  du. 

Integrating  a  second  time,  with  reference  to  w,  between  the 
limits  M=0  and  u=a,  we  shall  obtain  for  the  moment  of 
elasticity  «, 

«=y_Aa&' (143  w). 

Hence  it  follows  that  the  resistance  to  flexure  opposed  by  a 
solid  whose  transverse  section  is  rectangular,  will  be  propor- 
tional to  the  breadth  and  the  cube  of  the  depth. 

T> 

323   If  we  replace  A  in  this  expression  by  —,  we  shall 


RESISTANCE    OF   SOLIDS.  159 

obtain  the  moment  of  rupture  $  of  the  rectangle  ;  and  since 
V  is  in  the  present  case  equal  to  16,  we  shall  have 

i3=JBa6= (143  o). 

Thus  the  resistance  to  fracture  is  proportional  to  the  breadth 
and  the  square  of  the  depth. 

324.  If  the  solid  be  disposed  in  such  manner  that  the 
dimension  a  shall  become  vertical,  and  the  dimension  b  hori- 
zontal, the  expressions  for  the  moments  of  elasticity  and  rup- 
ture will  become  respectively 

and  by  comparing  these  expressions  with  those  obtained, 
when  the  dimension  a  was  supposed  horizontal,  we  shall 
deduce  the  proportions 

«:»'::  ab^  :  ba^  :  :  b*  :  a*, 

/3  :  /3'  :  :  ai^  :  ba'  :  :  b  :  a. 
It  thus  appears  that  the  resistance  to  flexure  when  the  broader 
face  b  is  placed  vertically,  will  be  to  that  exerted  when  the 
narrower  face  a  is  vertical,  as  the  square  of  the  broader  face  to 
the  square  of  the  narrower.  But  that  the  resistances  to  frac- 
ture in  similar  cases  are  proportional  simply  to  the  first  powers 
of  the  same  quantities. 

325.  If  in  the  expressions  (143  n)  and  (143  0),  we  make 
az=b,  we  shall  obtain  for  the  moments  of  elasticity  and  rup- 
ture of  a  prism  with  a  square  base, 

«=^TVAa%     0=lBa^ (143  jo). 

326.  Let  the  transverse  section  of  the  solid  be  a  rhombus, 
{Fig.  143),  whose  diagonals  are  represented  by  2p  and  2q, 
and  let  the  diagonal  2q  be  placed  vertically.  If  we  first 
determine  the  moment  of  elasticity  of  the  triangle  oBG,  that 
of  the  rhombus  can  be  immediately  deduced  by  simply  multi- 
plying by  the  number  2.  The  limits  between  which  the  first 
integration  with  reference  to  the  variable  v=ow,  should  be 
effected,  are  -«=0,  and  v=ot.  But  from  the  similarity  of  tri- 
angles, we  have  the  proportion 

ao  :  ot  :  :  oD  :  DC, 
or, 

u:  ot  ::p  :  q] 


100  STATICS. 

whence, 

ot=—  ; 
P 
and  the  limits  of  the  first  integration  will  therefore  be  v=o, 

and  v=—.     Making  these  substitutions  in  the  general  for- 
mula for  the  moment  of  elasticity,  we  shall  obtain 

2Af''duf"^v'>dv=2Af"duX^(^-^)  '=tA^'  f\=du. 
t/ot/o  «/o  \p  /  ]J^»/  0 

Integrating  a  second  time,  with  reference  to  the  variable  u, 

between  the  limits  71  =  0,  and  u=p,  the  moment  of  elasticity 

of  the  triangle  aBC  becomes 

and  by  doubling  this  expression,  we  find  for  the  moment  of 
elasticity  «  of  the  rhombus, 

T> 

327.  If  in  this  expression  we  replace  A  by  —,  we  shall  ob- 
tain the  value  of  the  moment  of  rupture  &,  which,  since 
Y=q,  will  become 

^  =  L-Xpq=  =  ^Bpq='. 

328.  If  we  make  2y=q,  the  rhombus  will  become  a  square, 
and  the  values  of  »  and  p  will  reduce  to 

or  if  the  side  of  the  square  be  denoted  by  a,  we  shall  have 
the  relation  a^=2p'',  and  therefore 

«=iAx^=-i-Aa*,  '5=iBx  2^  =-g-^  Bad- 
aud by  comparing  these  expressions  with  those  obtained 
(Art.  32o)  for  the  moments  of  elasticity  and  rupture  of  a 
prism  with  a  square  base,  when  the  sides  of  the  base  are 
respectively  vertical  and  horizontal,  we  shall  find  that  the 
resistance  to  flexure  will  be  the  same  whether  the  diago- 
nal or  side  of  the  square  be  disposed  vertically;  but  that 
the  resistance  to  fracture  when  the  side  is  vertical,  will  be 


RESISTANCE    OF    SOLIDS.  ^61 

greater  than  when  the  diagonal  is  vertical,  in  the  ratio  of 
v^(2)  to  1. 

329.  When  the  section  is  a  circle  whose  radius  is  equal  to 
r,  the  integration  with  reference  to  the  variable  v  must  be 
effected  between  the  limits  v=0,  and  v=^{2ru — u");  and 
the  second  integration, with  reference  to  u,  between  the  limits 
u=0,  and  u=2r.  Thus,  the  expression  for  the  moment  of 
elasticity  will  be 

«=2A/     duj  v^dv  =  \kj ^   {2ru—u''ydu..{U3q). 

For  the  purpose  of  effecting  the  second  integration,  we 
make  r—u—z,  which  gives 

du=  —  dz,     2ru — w^  =r^  — z^. 
Substituting  these  values  in  the  expression  for  «,  and  observ- 
ing that  the  limits  u=0,  and  ?«=2r,  correspond  to  the  values 
z=-{-r,  and  z=—r,  we  shall  obtain 

{2ru—u^ydu=—J  ^  {r'—z^ydz  = 
-f[^{r^  —z^  )(r=  -z^  fdz  \ 


or, 


r  i^ru—u''  y  du  =J    z^  {r'  —z'  fdz 

-J^j'ir^  -z^Ydz (143  r). 

The  first  term  of  the  second  member,  being  integrated  by 
parts,  gives 

f'z^  (r«  —z^fdz^-r^r^  —z^Y2zdz= 
-{r^-z^f  .'^^■^\fl{r^-z^fdz (143  s). 

3Z 

The  quantity  {r^—z")"'^  will  reduce  to  zero,  when  z=-f-r, 
or  z=—rt  this  term  will  therefore  disappear;  and  the  last 
term  being  resolved  into  factors  will  reduce  equation  (143  s) 
to 

[[z^r^  —z^fdz=lj*jr*  -z^fr^  —\J_J^''  -z^Vz'dz'r 
whence,  by  transposition  and  reduction,  we  obtain 


16^  STATICS. 

This  value  being  substituted  in  (143  r)  gives 

J    \2ru—u''ydu—  —  lr^j\r^  —z'^fdz. 

But  the  integral  /     (7'^  —z^ydz  represents   the  area  of  a 


«/  0 


semicircle  whose  radius  is  equal  to  r.     This  area  being  ex- 
pressed by  ^?rr",  we  shall  have 

•2r 

{2ru—u^ydu=—^^^\ 

and  by  substituting  this  value  in  the  expression  (143  q)  for 
the  moment  of  elasticity  «,  it  will  become 

330.  To  determine  the  moment  of  rupture  j8,  we  replace 

A  by  —  or  —,  and  thus  obtain 
V         r 

331.  By  comparing  these  values  with  the  expressions 
(143  jo),we  shall  find  that  the  moments  of  elasticity  and  rup- 
ture  of  a  square  are  to  those  of  the   inscribed  circle  as 

1  ">  Î6- 

332.  The  moment  of  elasticity  of  a  tube  or  hollow  cylin- 
der whose  exterior  and  interior  diameters  are  represented  by 
r'  and  r",  will  be  determined  by  taking  the  difference  of  the 
moments  of  the  exterior  and  interior  sections.  Thus  we 
shall  have 

«=iA5r(r"'— r"*), 
and  the  moment  of  rupture  /3  will  be  found  by  replacing  A  by 

B       B     , 

^or~;  hence, 

(8-iBT ~ — 

r' 

333.  If  the  section  of  the  hollow  cylinder  be  supposed 
equal tothat  of  a  solid  cylinder,  the  radius  of  the  latter  being 
denoted  by  r,  we  shall  have  the  relation 

f2  ^=,f'2  y>"3. 


RESISTANCE    OP   SOLIDS.  163 

and  the  resistances  to  fracture  opposed  by  the  two  will  be  to 
each  other  as 

J—  :  {r'^-— r"^y, or  as  ~^~  :  {r'^—r'"')'  ; 

replacing  r'^  _ r"^  by  its  value  r^,  this  ratio  will  be  reduced  to 

r 
The  first  term  of  this  ratio  must  always  exceed  the  second  : 
thus  the  resistance  to  fracture  opposed  by  the  hollow  cylinder 
will  always  be  greater  than  that  offered  by  the  solid  cylinder  ; 
and  since  the  value  of  the  first  term  may  be  increased  indefi- 
nitely without  affecting  that  of  the  second,  it  follows  that 
the  resistance  of  the  hollow  cylinder  may  likewise  be  in- 
creased indefinitely  without  changing  the  area  of  its  section. 
334.  Let  a  and  b  represent  the  breadth  and  height  of  a 
rectangle  inscribed  in  a  circle  whose  diameter  is  denoted  by 
D  :  we  shall  have  the  relation  a=^  -|-6=^  =0^  ;  and  therefore, 
«62=«^D2— «3). 

But  the  moment  of  rupture  of  a  rectangle  being  proportional 
to  the  breadth  and  the  square  of  the  depth  (Art.  323),  if  we 
wish  the  resistance  to  fracture  to  be  a  maximum,  we  must 
differentiate  the  preceding  expression  with  reference  to  a,  and 
place  the  first  differential  coefficient  equal  to  zero  :  we  shall 
thus  obtain 


da 
and  therefore, 


:I)'-—3a'=0: 


Hence,  the  strongest  rectangular  solid  which  can  be  cut  from 
a  given  cylinder  will  be  that  in  which  the  diameter  of  the 
cylinder,  the  depth  of  the  rectangular  section,  and  its  breadth, 
shall  be  to  each  other  as  the  square  roots  of  the  numbers  3, 
2,  and  1. 

L2 


16^  STATICS. 


Of  the  Figure  of  the  /Solid  after  Flexure. 

335.  We  will  now  consider  the  form  of  the  curve  Aiiu'B 
{Fig.  140)  assumed  by  the  fibres  whose  lengths  remain  inva- 
riable. For  this  purpose,  let  AM  {Fig.  144)  represent  the 
solid  which  is  firmly  fixed  at  its  extremity  A,  and  subjected 
to  the  action  of  the  weight  P,  applied  at  the  other  extremity, 
in  a  direction  perpendicular  to  the  original  direction  of  the 
axis  of  the  solid.  Then  denoting  by  «  the  moment  of  elas- 
ticity, the  equation  (143  i),  which  expresses  a  condition  of 
equilibrium,  when  the  solid  merely  undergoes  flexure,  with- 
out being  ruptured,  will  become 

or  by  substituting  for  the  radius  of  curvature  R  its  general 

value  J '^    ,  this  equation  will  reduce  to 

_d^ 

d^ 


(^-'1^) 


336.  In  Uke  manner,  when  the  solid  is  about  to  be  rup- 
tured, if  we  substitute  /?  for  the  moment  of  rupture,  in  equa- 
tion (143  m),  we  shall  obtain 

li=V{x'—x) (143  w). 

337.  Let  c  denote  the  horizontal  distance  AB  between  the 

extremities  of  the  solid, 
/,  the  ordinate  BM, 
s,  the  length  of  the  arc  AmM^ 

»,  the  angle  included  between  the  tangent  to  the  curve 
at  the  point  M  and  the  horizontal  line. 
Then,  since  the  curvature  is  supposed  to  be  extremely  small, 
even  at  the  instant  when  fracture  takes  place,  the  expression 

-^,  which  represents  the  tangent  of  the  angle  formed  by  the 


RESISTANCE    OF   SOLIDS.  165 

element  of  the  curve  with  the  axis  of  x,  will  also  be  extremely- 
small,  and  its  square  may  therefore  be  neglected  in  compari- 
son with  unity.     Thus  the  equation  (143 1)  will  be  reduced  to 

Multiplying  by  dx,  we  obtain 

»--^dx=Vlc — x)dx  ; 
dx^  ^ 

and  by  integration,  we  have 

-l=K— Ï) (^*^''>- 

The  arbitrary  constant  introduced  by  integration  is  equal  to 

zero;  since,  when  ar=0,  —^,  which  represents  the  tangent  of  the 
dx 

angle  included  between  the  element  of  the  curve  and  the  axis 

of  abscisses,  is  likewise  equal  to  zero. 

Multiplying  again  by  dx  we  have 

»-~dx~Vl  ex — ^  \dx: 
dx  V  27       ' 

and  performing  a  second  integration,  there  results 

the  constant  will  be  equal  to  zero,  since  x—0  gives  y=0. 

338.  If  in  this  expression  we  make  x—c,  the  ordinate  y 
will  become  equal  to/;  hence  we  shall  have 

^-(2-6)=.-^  3 ("^">- 

In  like  manner,  by  making  x—c  in  equation  (143  v),  we  shall 

have  '^=  tang  <»,  and  therefore 
dx 

P/  ,     c2\     P    ça 

P  3/* 

or,  replacing  —  by  its  value  —  deduced  from  the  preceding 

equation,  we  have 

3/- 
tang  «=  ;j- (143  x). 


i6ê  STATICS. 

339.  To  determine  the  length  s  of  the  arc  A^nM,  we  take 
the  general  expression  for  the  element  ds  of  this  arc, 

which,  being  developed,  rejecting  all  but  the  two  first  terms 
as  inconsiderable,  gives 

and  by  replacing  — ^  by  its  value   (143  v),  this  equation 
ax 

becomes 

pa 

Integrating,  we  obtain 

F-  {c^x^     cx'^  ,  x'\ 

and  by  making  x=c,  the  value  of  the  entire  arc  AwiM  becomes 

pa   /^6       f.s       c5v  pa       c« 

^='=+^i6-8+ro)='^+^^r5' 

pa 

or,  replacing  — by  its  value  deduced  from  equation  (143  w\ 

this  expression  reduces  to 

5=c  +  |: (143  2/). 

5c 

340.  When  the  weight  P  is  just  sufficient  to  fracture  the 
solid,  the  rupture  will  take  place  at  the  supported  end  ;  since 
the  moment  V{c—x)  of  the  force  P  will  be  the  greatest  when 
a:=0:  the  equation  (143  ii)  will  then  become 

/3=Pc (143  z)  ; 

diJ^ 
or,  if  the  curvature  be  still  supposed  so  small  that  — -  may  be 

neglected  in  comparison  with  unity,  the  equation  of  the  curve 
will  be  the  same  as  when  the  flexure  was  extremely  slight, 
and  we  shall  therefore  have 

/3 


P= 


3/^ 
^      5c 


341.  Let  it  now  be  supposed  that  the  solid  is  loaded  with 


RESISTANCE    OF   SOLIDS.  167 

weights  distributed  uniformly  throughout  its  length.  Denote 
by  z  the  absciss  of  any  point  between  M  and  wi,  and  by  p  the 
weight  supported  by  a  portion  of  the  solid  which  corresponds 
to  a  unit  of  length  of  the  absciss  :  then  since  the  distribution 
of  the  weights  is  supposed  uniform,  we  shall  have  the 
proportion 

\:  J)  w  dz  '.  pdzj 

the  weight  supported  by  the  element  of  the  solid  whose  pro- 
jection on  the  axis  of  x  is  represented  by  dz.  The  moment 
of  this  weight,  with  reference  to  the  point  w,  will  be 
pdz{z — x),  and  the  sum  of  the  moments  of  all  the  weights 
supported  between  M  and  m,  taken  with  reference  to  the 
same  point  m,  will  be 

.fp{z — x)dz. 
This  integral  should  be  taken  between  the  limits  z=c  and 
z=x,  the  quantity  x  being  regarded  as  invariable  :  thus  we 
shall  have 


•/  X 


q2  ^2 

p{z—x)dz=p — ~ px{c—x) (143  a'). 


2 

But  the  condition  of  equilibrium  requires  that  the  sum  of 

d^V 
these  moments  shall  be  equal  to  «  -^,  the  sum  of  the  mo- 

dx'' 

ments  of  the  resistances  offered  by  the  several  fibres.     Hence, 

we  obtain 


«-^  =p  (t-^\  —px{c—x)  =  i/?c3  —pcx+lpx^ . 
d  integrating,  we  obtaii 


Multiplying  by  dx,  and  integrating,  we  obtain 

dy_ 

dx 

and  multiplying  a  second  time  by  dx^  and  integrating,  there 
results 

»y=p{\c^x^  —\cx^-[-yc'). 

Making  x=c,  y=f-,  and  -^=tang  <v,  we  find 

/=^(ic«-ic*+ic^)=f  .'^ (143  h% 

tang  «=%c»-ic3+ic=')=|^. 


1G8  STATICS. 

342.  When  the  weights  distributed  along  the  soHd  are  just 
capable  of  producing  rupture,  the  fracture  will  take  place  at 
the  supported  end,  since  the  expression  (143  a')  which  repre- 
sents the  sum  of  the  moments  of  these  weights  will  evidently 
be  the  greatest  when  x~0.  This  sum  being  then  equal  to 
the  moment  of  rupture  /3,  we  shall  have 

2/3 

/3  =  i;.c^     cp  =  -j (143  c'); 

the  expression  pc  represents  the  entire  weight  distributed 
along  the  solid. 

343.  If  we  make  cp=V,  and  compare  the  values  (143  6') 
and  (143  w)  of  the  ordinate  /,  it  will  appear  that  the  depres- 
sion of  the  point  M  below  the  horizontal  line  Ax,  produced 
by  the  action  of  the  weight  P  applied  at  the  point  M,  will  be 
greater  than  the  depression  produced  by  an  equal  weight 
distributed  uniformly  along  the  solid,  in  the  ratio  of  8  to  3. 

And  by  comparing  the  values  of  —  in  equations  (143  c')  and 

(143  ;r)  we  shall  perceive  that  the  weiglit  necessary  to  frac- 
ture the  solid,  when  distributed  uniformly,  will  be  double  that 
required  when  it  is  applied  at  the  extremity  M. 

344.  Tt  frequently  occurs  that  the  weight  of  the  solid  forms 
an  important  part  of  the  load  which  it  is  required  to  sustain. 
The  eiFect  produced  by  this  weight  is  readily  calculated  by 
regarding  it  as  uniformly  distributed  throughout  the  solid. 
Thus,  if  the  solid  be  loaded  with  its  own  weight  V=pc,  and 
a  weight  P  applied  at  its  extremity  M,  the  sum  of  the 
moments  of  the  weight  P,  and  the  weight  of  that  portion  of 
the  solid  which  lies  to  the  right  of  the  point  w,  taken  with 
reference  to  that  point,  will,  by  Arts.  337  and  341,  be 

P(c— :r)  +p{l  c2  — ca.-  +  |.r2)  ; 
and  in  case  of  equilibrium,  we  shall  have 

«^=P(c-^)+Kic=>-c.r+i^-^) (143  <^'); 

or,  if  the  solid  be  supposed  on  the  point  of  being  ruptured, 
the  fracture  taking  place  at  the  point  A,  for  which  a;=0,  the 
condition  of  equilibrium  will  be 

/3=Pc-f  ipc^». 


RESISTANCE    OF    SOLIDS.  169 

345.  The  expression  (143  d')  gives,  by  two  successive 
integrations, 

and  by  making  x=Cj  y=/,  and  ~^=  tang  «,  we  obtain 

tano-a;=— (iP  +  i«c')=  ^P+P'    4/ 

,5=c(P+i;^c)  =  c(P+iF) 

346.  When  the  soHd  is  supported  in  a  horizontal  position 
at  its  two  extremities  M  and  M'  (Fig:  145),  and  loaded  with 
weights  at  its  middle  point  A,  the  results  obtained  Arts.  337- 
340  will  apply  to  each  half  of  the  curve  assumed  by  the 
solid  ;  for  we  may  regard  either  half  as  perfectly  immoveable, 
and  suppose  the  other  portion  to  be  solicited  by  a  force  acting 
at  its  extremity  and  equal  to  the  resistance  offered  by  one  of 
the  points  of  support.     Hence,  if  we  denote  by 

2P5  the  weight  suspended  at  the  middle  point, 
2c,  the  distance  between  the  points  of  support, 
25,  the  length  of  the  curve, 
/,     the  sagitta  CA, 

a,    the  angle  included  between  the  line  MM'  and  the 
tangent  to  the  curve  at  M  or  M'  ; 
the  resistance  exerted  by  each  fixed  point  in  the  vertical 
direction  will  be  equal  to  P,  one-half  the  weight  applied  at 
A,  and  the  formulas  (143  w),  (143  x),  (143  y),  and  (143  z)  will 
become  immediately  applicable  to  the  present  case.     Hence, 
P    c2j2c^  2P 
•^     «'3        «    -48 (^"^^^h 

tang  .=g 

25=2c+^^ 
oc 

/î=cP (143/). 

15 


170  STATICS. 

The  value  of  /  indicates  that  the  depression  of  the  soHd  at 
tlie  middle  point,  or  the  sagitta  AC,  will  be  proportional  to  the 
weight  2P,  and  the  cube  of  the  distance  between  the  points 
of  support. 

347.  The  expressions  deduced  in  the  preceding  article  have 
been  obtained  upon  the  supposition  that  the  resistances  op- 
posed by  the  fixed  points  were  exerted  in  a  vertical  direction  ; 
whereas,  the  resistance  is  actually  exerted  in  the  direction 
of  the  normal  to  the  curve  at  the  point  M  or  M'  ;  and  in 
some  instances  the  inclination  of  this  normal  to  the  vertical 
line  is  too  great  to  be  neglected.  This  circumstance  will 
seldom  occur  except  in  the  case  of  fracture,  the  curvature 
of  the  solid  being  then  greater  than  in  the  case  of  a  mere 
flexure.  If  we  represent  the  resistance  exerted  at  M'  by  the 
hne  M'F,  and  resolve  this  force  into  two  components  which 
shall  be  respectively  vertical  and  horizontal,  the  latter  com- 
ponent ME  will  be  equal  and  opposite  to  the  similar  compo- 
nent of  the  resistance  at  the  point  M,  and  the  vertical 
component  M'D  will  be  equal  to  P,  or  to  one-half  the  weight 
supported  at  the  ?Tiiddle  point  of  the  solid.  The  value  of 
the  horizontal  component  M'E  may  be  readily  found  ;  for  we 

have 

M'E =DF=M'DX tang  DM'F=P  .tang«. 

When  the  equilibrium  subsists,  and  the  solid  is  on  the  point 
of  being  ruptured,  the  moment  of  rupture  must  be  equal  to 
the  sum  of  the  moments  of  the  vertical  and  horizontal  com- 
ponents. The  moment  of  the  former,  with  reference  to  the 
point  A,  has  been  found  equal  to  cP  ;  that  of  the  latter  will 
obviously  be  P  tang  a  xAC=P  tango- ,/;  thus,  the  con- 
ditions of  equilibrium  will  become 

j8=Pc  +  P  tang*./; 

or,  if  we  suppose  the  curve  to  be  represented  by  the  same 

3f 
equation  as  in  Art.  337,  in  which  case  tang*'=^,  this  rela- 
tion may  be  written 

348.  If  the  weight  be  uniformly  distributed  throughout  the 


RESISTANCE    OF    SOLIDS.  171 

length  of  the  solid,  we  may  regard  each  half  as  firmly  fixed 
at  the  point  A,  and  solicited  at  the  same  time  by  a  system  of 
parallel  forces  applied  at  every  point  of  the  solid,  and  acting 
downwards  ;  and  by  a  single  force  equal  to  their  sum,  or  to 
the  resistance  offered  by  the  point  of  support,  applied  at  the 
extremity  of  the  solid,  and  acting  upwards.  Thus,  the  case 
will  be  the  same  as  that  considered  in  Art.  344,  with  the  ex- 
ception that  the  forces  arising  from  the  weights  uniformly 
distributed  along  the  solid  are  exerted  in  contrary  directions. 
The  equations  obtained  in  that  case  will  therefore  become 
applicable  to  the  present  one  by  simply  changing  the  signs 
of  the  moments  of  these  forces,  and  replacing  P  by  pc  ;  we 
shall  thus  obtain 

a,^=,Cp{cX  —  \X')—p{\c^X  —  \cX^-{-\x'^) {^^'^  g')'\ 

ay—cp{\cx''—\x^)—p{\c''x''—\cx''-\-^^x^)  .....  (143 /i'); 
ç>—cp  .  c — cp  .\c (143  Ï)  ; 

making  x=c^  y=-fi  -f—  tang  a»,  we  obtain 


^■x  C»=- 


24' 


tang.  =  ^(l-i-i+^-i)c3=i?Ç=|', 

^=cp.\c (143/:'). 

By  comparing  this  value  of/  with  that  obtained  in  equa- 
tion (143  e'),  it  will  appear  that  the  depression  of  the  solid  at 
its  middle  point  produced  by  a  weight  2pc  uniformly  dis- 
tributed throughout  the  solid,  will  be  less  than  that  produced 
by  the  same  weight  suspended  at  the  middle  point,  in  the 

ratio  of  5  to  8.     And  by  comparing  the  values  of  —  given  by 

equations  (143  k')  and  (143/')  we  shall  perceive  that  the  solid 
will  be  equally  liable  to  fracture  by  the  action  of  the  weight 
2pc  distributed  uniformly,  or  by  half  that  weight  applied  at 
its  middle  point. 

349.  The  preceding  expressions,  like  those  in  Art.  346, 
have  been  obtained  upon  the  supposition  that  the  resistances 
offered  by  the  fixed  points  are  exerted  in  vertical  directions. 


172  STATICS. 

In  the  case  of  riiplure,  the  hne  of  direction  of  the  resist- 
ance may  deviate  so  far  from  the  vertical  as  to  render  the 
above  supposition  inadmissible.  We  then  resolve  this  resist- 
ance, as  in  Art.  347,  into  two  components  respectively  vertical 
and  horizontal  ;  the  former  will  be  represented  by  pCy  and  the 
latter  by  jjc  •  tang  a.  In  case  of  equilibrium,  it  will  simply 
be  necessary  to  add  to  the  second  member  of  equation  (143  i') 
the  moment  jjc  .  tang  «  X/,  of  the  horizontal  component;  thus, 
we  shall  have 

^  =  cp  .c—cp.  Ic-jrcp  .  tang  «  ./=rc/v(|e+/tang<v), 

Sf 
or,  by  replacing  tango;  by  its  value  p^,  we  have 

oc 


and  therefore. 


4^ 


2cp=- 


(-1") 


we  here  suppose  that  the  equation  of  the  curve  remains  the 
same  as  in  Art.  337. 

350.  If  the  solid  be  loaded  at  the  same  time  with  a  weight 
2P  at  its  middle  point,  and  its  own  weight  2pc=z2F'  uni- 
formly distributed,  the  case  will  be  similar  to  that  considered 
in  the  two  preceding  articles,  with  the  exception  that  the 
force  applied  at  the  extremity  of  the  solid  will  now  be  repre- 
sented by  P+/>c=P+P':  thus,  when  we  suppose  the  resist- 
ances exerted  by  the  fixed  points  to  act  vertically,  we  shall 
obtain,  by  substituting  P-f-i>c  for  pc  in  the  first  terms  of  the 
second  members  of  equations  (143  h')  and  (143  i'), 

«y=(P+pc)(ic.r2  —}z-^)—p{lc^x-  —}ca;^+^\x*), 

^  =  (P+pc)c—cp .  ic (143  r)  ; 

which  give,  by  making  a:=c,  y=f,  and  pc=P', 

351.  But,  if  regard  be  had  to  the  oblique  direction  of 
the  resistance,  as  may  be  necessary  in  the  case  of  rupture, 
we  must   add  the  moment  of   the   horizontal   component 


RESISTANCE    OP   SOLIDS.  173 

to  the  second  member  of  equation  (143 1%  which  thus 
becomes 

^=(P+2?c)c— cp .  ic+(P+pc)  tang  a, ./; 
and  therefore, 

gp_2^-F(c+2/tang^)^ 
c-\-f  tang  a 
The  equation  (143  g')  likewise  gives,  by  replacing  pc  in  the 
first   term  of  the  second  member  by  P+^c,  and  making 

|=tang  », 

3P  +  2P'  4/ 

and  this  value  of  tang  a  may  be  regarded  as  sensibly  equal 
to  that  employed  in  the  preceding  expression  for  the  value 
of2P. 

352,  To  apply  the  several  results  which  have  been  ob- 
tained to  particular  cases,  it  will  be  necessary  to  substitute 
the  values  of  the  moments  of  rupture  and  elasticity  apper- 
taining to  the  figure  of  the  transverse  section.  We  must 
likewise  assign  to  A  and  B  the  coefficients  of  elasticity  and 
tenacity,  their  particular  values  which  depend  upon  the  nature 
of  the  substance,  and  which  are  supposed  to  have  been  pre- 
viously determined  by  experiment, 

353.  The  best  method  of  determining  the  values  of  A  and 
B  consists  in  supporting  a  prismatic  solid  at  its  two  ex- 
tremities in  a  horizontal  position,  loading  it  with  weights  at 
its  middle  point,  and  observing  the  sagittas  which  correspond 
to  different  weights  ;  or  simply,  the  weight  and  sagitta  at  the 
instant  when  the  fracture  is  about  to  take  place. 

If  the  transverse  section  of  the  solid  be  a  rectangle, 
whose  breadth  and  height  are  denoted  respectively  by  a  and 
6,  we  shall  have  (Arts.  322  and  323), 

et=j\kab'',     ^=iBa62  ; 
and  if  we  neglect  the  weight  of  the  solid  (Arts.  346  and  347), 

and  by  eliminating  «  and  /3,  we  obtain,  for  the  case  of  simple 
flexure, 


174  STATICS. 

/=2P-^?^,     orA=2pi?$4. (143^0; 

and  for  that  of  fracture 

«=^Pa^O+i?) ("=>"')' 

2c  being  the  interval  between  the  supports,  and  2P  the  weight 
with  which  the  solid  is  loaded. 

The  values  of  A  and  B  are  thus  expressed  in  functions  of 
quantities  which  are  readily  determined  by  observation. 

354.  If  the  weight  of  the  solid  2P'  be  likewise  taken  into 
consideration,  it  will  simply  be  necessary  (Art.  350)  to  add 
1 .  2P'  to  2P  in  equation  (143  m%  and  to  replace  equation 
(143  n')  by  the  formulas  of  Art.  351  :  we  shall  thus  have,  i» 
the  case  of  flexure, 

/=(2P  +  |.2P')-i?^,     A=(2P  +  f.2F)-^^')' 


and  for  that  of  fracture, 

6g  _(2P+2FXc+/.tang<^)— P^c 

3P  +  2P'   4/ 

*""^^=8P+5P'-T 
355.  If  the  solid  be  loaded  with  a  weight  2Q,,  and  if  the 
corresponding  sagitta  be  denoted  by  f,  we  shall  obtain  a 
value  for/'  similar  to  that  of /in  the  preceding  article  :  thus 
"we  shall  have, 

and  by  taking  the  difference  between  /  and  /',  the  weight  of 
the  solid  2P'  will  disappear,  and  we  shall  obtain 

/'_/=(2a-2P)_^,  A=(2a-2P)  — (^^^' 


4Aa63'  ^  -^          '  4.abHf'—f) 

Thus,  it  will  only  be  necessary  to  observe  the  increase  /'— / 
in  the  sagitta,  which  corresponds  to  a  given  increase  2Q,— 2P 
in  the  weights  suspended  at  the  middle  point. 


RESISTANCE   OF  SOLIDS.  175 


Of  Solids  of  equal  Resistance. 

356.  When  a  solid  having  the  prismatic  form  is  subjected 
to  an  eflbrt  which  tends  to  break  it,  there  will  always  be  a 
particular  point  at  which  the  fracture  will  be  most  likely  to 
take  place.  For,  the  moment  of  rupture  will  be  the  same  at 
every  point,  whilst  the  moment  of  the  force  applied  will  de- 
pend upon  its  distance  from  the  point  with  reference  to  which 
the  moments  are  taken.  Hence,  if  the  strength  of  the  solid 
be  sufficient  at  that  point  where  a  rupture  is  most  likely  to 
occur,  it  will  be  unnecessarily  great  at  other  points. 

357.  It  becomes  an  object,  therefore,  to  determine  the 
figure  of  the  solid  which  shall  be  uniformly  strong  through- 
out, since  the  adoption  of  such  a  figure  may  frequently  effect 
a  material  reduction  in  the  quantity  of  materials  employed. 
Solids  having  such  figures  are  called  solids  of  equal  resist- 
ance. 

358.  As  an  example,  let  a  body  ABM  {Fig.  146),  whose 
upper  surface  AB  is  horizontal,  and.  whose  two  lateral  faces 
are  vertical,  be  firmly  fixed  at  its  extremity  A,  and  subjected 
to  the  action  of  a  weight  P  suspended  from  its  other  extrem- 
ity. It  is  required  to  determine  the  form  of  the  under  surface 
BmM  such  that  the  solid  may  be  equally  strong  throughout, 
or  that  the  moment  of  the  weight  P  taken  with  reference  to 
any  point  in  the  length  of  the  solid,  shall  be  equal  to  the  mo- 
ment of  rupture  of  the  transverse  section  at  the  same  point. 

Denote  by  a  the  breadth  of  the  solid,  h  the  height  AM,  c 
the  length  AB,  x  the  variabl-e  absciss  Bjo,  and  v  the  corre- 
sponding ordinate  jmi  :  the  moment  of  rupture  of  the  section 

ah^ 
AM  will  be  (Art.  323)  B— -  ;  and  since  this  must  be  equal  to 

the  moment  of  the  force  P,  we  shall  have 

In  like  manner,  the  moment  of  rupture  of  the  section  pm 

av^ 
will  be  B-^ ,  and  the  moment  of  the  force  P  with  reference 
6 

to  a  point  in  this  section  will  be  Vx.     These  moments  being 


176  STATICS. 

equal  by  the  conditions  of  the  problem,  the  general  relation 
between  the  quantities  v  and  a:  will  become 

P:r=B-— ,     v^  =  — . 

D  C 

This  equation  evidently  appertains  to  a  parabola,  the  axis  of 
which  will  be  the  line  AB. 

359.  To  determine  the  figure  of  the  curve  assumed  by  the 
solid  when  bent,  we  observe  that  the  moment  of  elasticity  of 

the  section  p?n  will  be  (Art.  322)  A— —  =A —,    Hence,  if 

^^  12c^ 

y  denote  the  ordinate  of  the  curve  of  flexure  corresponding  to 
the  absciss  Ap=c—x,  the  conditions  of  equilibrium  in  case 
of  flexure  will  be  (Art.  337) 

A X  — 4=P^. 

Performing  two  successive  integrations,  and  remarking  that 

when  x—c^  -/-=0,  and  y=0,  we  obtain 
ax 

and  by  making  a;=0,  and  y=/,  we  find,  for  the  depression  of 
the  extreme  point  B, 

P    8c^ 
''~'A.'ab^' 
By  comparing  this  expression  with  that  obtained  in  equa- 
tion (143  w),  it  will  appear  that  the  depression / is  twice  as 
great  in  the  present  instance  as  when  the  solid  had  the  pris- 
matic form. 

360.  If  the  weight  supported  by  the  solid  be  distributed 
uniformly  along  its  length,  each  unit  of  length  being  sup- 
posed to  support  a  weight  jh  '^he  sum  of  the  moments  of 
these  weights,  taken  with  reference  to  the  point  A,  will  be 
(Art.  342)  pc.ic;  and  the  condition  of  equilibrium  will 
therefore  be 

B-—=pc.\c. 

0 


PRINCIPLE    OF   VIRTUAL    VELOCITIES.  177 

In  like  manner,  the  sum  of  the  moments  of  the  weights  sup- 
ported between  the  points  p  and  B,  taken  with  reference  to 
the  point  /?,  will  be  fx .  \x.     Hence,  we  shall  have 
„«v^  ,  hx 

b      ^      ^  c 

the  equation  of  a  right  line. 

361.  The  preceding  examples  will  be  sufficient  to  illus- 
trate the  manner  in  which  the  form  of  the  solid  of  equal  re- 
sistance may  be  determined  when  the  distribution  of  the  load 
is  previously  known. 


Of  ike  Principle  of -Virtual  Velocities. 

362.  The  principle  of  virtual  velocities,  which  was  dis- 
covered by  Galileo,  and  very  fully  developed  by  John 
Bernouilli  and  Lagrange,  may  frequently  prove  of  great 
utility  in  stating  the  analytical  conditions  of  statical  problems. 
Indeed,  it  is  regarded  by  Lagrange,  who  has  adopted  it  as  the 
basis  of  his  "Mécanique  Analytique,"  as  so  essential,  that  he 
considers  all  the  general  methods  which  can  be  employed  in 
the  solution  of  questions  relating  to  equilibrium,  as  being 
nothing  more  than  applications  more  or  less  direct  of  this 
general  principle. 

363.  A  virtual  velocity  is  the  path  described  by  the  point 
of  application  of  a  force,  when  the  equilibrium  is  disturbed 
in  an  infinitely  small  degree.  Thus,  by  supposing  that  the 
point  of  application  vi  of  a  force  P  {Fig.  147)  is,  by  an 
instantaneous  derangement  of  the  system,  transferred  to  ??, 
the  small  line  mn  which  it  describes  is  called  the  virtual 
velocity  of  the  point  m. 

364.  If  this  virtual  velocity  be  projected  upon  the  direction 
of  the  force,  it  will  occupy  thereon  the  small  space  ma,  and 
the  product  of  the  force  P  by  this  projection  ma  is  called  the 
moment  of  this  virtual  velocity,  or,  sometimes,  the  moment 
of  the  force  ;  it  should  however  be  observed,  that  the  term 
moment  is  here  employed  with  a  very  different  signification 
from  that  usually  implied. 

The  principle  of  virtual  velocities,  as  will  be  demonstrated, 

M 


178  STATICS. 

consists  in  this,  that  when  the  system  is  in  equihbrio,  the  sum 
of  these  moments  is  equal  to  zero  ;  thus,  if  P,  P',  P",  &.C., 
represent  different  forces  appUed  to  a  system,  and  p,  p',  ;j", 
&c.,  the  projections  of  the  virtual  velocities  on  the  directions 
of  these  forces,  we  must  have  in  case  of  equilibrium, 

Fp  +  P'p'+F'Y +Ôcc.=0 (144). 

It  is  necessary  to  remark  that  when  any  one  of  these  pro- 
jections p,  p\  p'\  (fcc,  falls  upon  the  prolongation  mh 
{Fig.  148)  of  the  force  P,  applied  at  m,  this  projection  must 
be  regarded  as  negative  ;  and  since  the  forces  P,  P',  P",  <fcc., 
are  all  considered  as  having  the  positive  sign,  the  moment 
corresponding  to  this  negative  projection,  must  likewise  be 
affected  with  the  negative  sign  ;  thus,  the  equation  (144)  will 
express  that  the  algebraic  sum  of  the  moments  is  equal  to  zero. 

365.  This  principle  will  first  be  demonstrated  for  that  case 
in  which  the  forces  are  applied  to  a  single  point.  Let  P,  P', 
P",  &-C.,  represent  any  number  of  forces  applied  to  the  point 
m  {Fig.  149),  and  sustaining  it  in  equilibrio  ;  if,  by  the  effect 
of  an  infinitely  small  derangement,  the  point  rti  be  trans- 
ported to  «,  the  line  tnn  being  infinitely  small,  may  be 
regarded  as  a  right  line.  Let  the  axis  of  x  be  supposed  to 
coincide  in  direction  with  the  line  mn,  and  denote  by  «,  «',  <«", 
&c.,  the  angles  formed  by  the  several  forces  with  this  axis  ; 
we  shall  have,  since  an  equilibrium  subsists  in  the  system, 

Pcosa  +  P'  cos  a'+P"  cos4"  +  (kc.  =0: 
multiplying  the  several  terms  of  this  equation  by  the  line  W7i, 
which  will  be  denoted  by  z,  we  shall  obtain 

Vz  cos  «+PV  cos  *'-fF'z"  cos  «"+&c.  =0 (145). 

But  it  is  evident  that  z  cos  «,  or  mn .  cos  nml^  is  equal  to  the 
small  line  'ml,  the  projection  of  mn  on  the  direction  of  the 
force  P.  Thus  z  cos  «  represents  the  same  quantity  as  the 
letter  p  in  equation  (144).  The  same  remarks  being  appli- 
cable to  the  other  forces,  the  several  products  z  .  cos  cl,  z  cos  «", 
&c.,  may  be  replaced  by-  p\  p",  <fcc.,  the  projections  of  the 
virtual  velocity  of  the  point  771  upon  the  directions  of  these 
forces,  and  the  equation  (145)  will  then  become 

P;)+P>'+P>"+(fcc.=0  ; 
from  which  we  conclude  that  the  principle  of  virtual  velocities 
is  true  when  the  forces  are  applied  to  a  single  point. 


PRINCIPLE    OP    VIRTUAL    VELOCITIES,  179 

366.  The  most  general  case  of  this  principle  which  usually 
presents  itself,  is  that  in  which  the  several  forces  P,  P',  P", 
(fcc,  are  applied  to  different  points  of  a  body  or  system  of 
bodies  :  these  points  preserving  their  distances  invariable, 
may  be  regarded  as  connected  with  each  other  by  inflexible 
right  lines.  Before  examining  the  general  state  of  the  system 
when  the  equilibrium  has  been  slightly  disturbed,  we  will 
consider  singly  one  of  these  inflexible  right  lines  mw',  at  the 
instant  when  the  point  m  has  been  brought  into  the  position 
denoted  by  ?i.  The  other  extremity  m!  of  this  right  line 
will  at  the  same  time  change  its  position,  and  may  be  situated 
either  above  mm'  {Fig.  150),  or  beneath  it  {Fig.  151)  :  let 
it  be  first  supposed  above  mm',  and  the  line  7nm'  will  then 
assume  the  position  mi'  {Fig.  152)  :  the  lines  mn  and  on'n' 
may  be  regarded  as  infinitely  small  when  compared  with  the 
lines  min'  and  nn',  since  the  derangement  of  the  system  is 
supposed  infinitely  small.  If  the  points  m  and  n'  be  con- 
nected by  a  right  line  we  shall  form  a  triangle  m/m'n',  in  which 
the  side  m'n'  being  infinitely  small,  the  angle  n'mm'  will  like- 
wise be  infinitely  small,  and  the  arc  w'a,  which  measures  this 
angle,  may  therefore  be  regarded  as  a  right  line.  But  this 
arc  beinof  described  with  a  radius  ma,  if  we  assume  mb=zma 
{Fig.  153),  the  angle  bn'a  being  an  angle  in  a  semicircle,  will 
be  equal  to  a  right  angle,  and  may  be  considered  equal  to  the 
angle  mn'a.  For,  since  the  angle  ii'nia  is  infinitely  small,  the 
angle  m,7i'b  must  be  so  likewise,  and  the  angles  bn'a,  mn'a,  will 
therefore  differ  by  an  infinitely  small  quantity.  Thus,  the 
triangles  mn'a  and  ii'la  {Fig.  152)  being  right-angled  and 
having  a  common  angle  a,  will  be  similar,  and  we  shall  there- 
fore have  the  proportion 

m,a  :  n'a  :  :  n'a  :  la. 
But  n'a  being  infinitely  small  with  respect  to  ma,  la  must  be 
infinitely  small  with  respect  to  n'a  ;  and  since  n'a  is  an  in- 
finitely small  quantity  of  the  first  order,  la  will  be  one  of  the 
second  order.  Hence,  the  quantity  la  may  be  neglected,  and 
mrî  may  be  regarded  as  equal  to  ml  ;  thus  we  shall  have 
mn'=mm'-±m'l. 

In  a  similar  manner  may  it  be  proved  that  if  with  the  point  nf 

M2 


180  STATICS. 

as  a  centre,  and  radius  n'm,  we  describe  the  arc  ma',  we  shall 
obtain 

tmi'=7in'-^7ih, 
and  by  placing  these  values  of  m?t'  equal  to  each  other,  we  find 

vnn'  -\-  ni'l=7in'  +  nh  ; 
but  the  right  line  min'  being  supposed  inextensible,  it  must 
preserve  its   length  invariable  in   its  new  position  ;  hence, 
mm'—nn'  ;  and  by  suppressing  these  equal  terms  in  the  pre- 
ceding equation,  we  obtain 

'}n'l=nh. 
Again,  the  lines  Qnm'  and  nn'  form  with  each  other  an  infi- 
nitely small  angle  ;  for,  if  they  intersect  at  a  point  o  {Fig.  154), 
we  shall  have  a  triangle  m'on',  two  of  whose  sides  are  of  finite 
extent,  the  third  side  m'n'  being  infinitely  small  ;  thus,  the 
angle  o  will  likewise  be  infinitely  small.  It  results  from  the 
preceding  remarks,  that  if  the  perpendicular  iik  be  demitted 
on  the  side  oTim'  (Fig.  152)  we  shall  have 

nh=9nk] 
and  by  substituting  this  value  of  nh  in  the  preceding  equation, 
we  find 

'm'l=mk, 

which  proves  that  the  projections  mk  and  m'l  of  the  virtual 
velocities  mn  and  m'n'  of  the  points  m  and  m'  are  equal  to 
each  other. 

367.  Let  us  now  suppose  that  the  point  w  {Fig.  155)  is 
transported  to  n,  and  that  the  extremity  m'  falls  at  n'  below 
Tnm'.  It  may  be  proved  as  in  the  former  case,  that  the  angle 
0  is  infinitely  small,  and  consequently  that  the  projections  ol 
and  oh  may  be  regarded  as  equal  to  o?i'  and  07i  ;  whence, 

on'=om'-{-m'l,     on  =07n — mA  ; 
by  the  addition  of  these  equations,  we  obtain 

07i'-{-o7i—om'-\-om-\-m'l— mh  ; 
or, 

7in'  =mm'-\-m'l  —  mh  ; 

but  7in'  and  mtn'  are  equal  to  each  other,  and  therefore 

7n'l—7nh, 


PRINCIPLE    OP    VIRTUAL    VELOCITIES.  181 

which  proves  that  the  projections  of  the  virtual  velocities  are 
still  equal. 

368.  In  this  demonstration  it  has  been  supposed  that  the 
derangement  of  the  system  is  such  as  to  preserve  the  lines 
mm'  and  nn'  in  the  same  plane.  This  restriction  is  however 
entirely  unnecessary.  For,if  we  suppose  that  mm'  and  nn'^re 
not  contained  in  the  same  plane,  we  can  draw  through  the 
points  n  and  7t'  {Fig.  152)  planes  perpendicular  to  the  line 
mm',  intersecting  this  line  at  the  points  k  and  I,  the  projections 
of  n  and  n'.  Then,  if  a  line  be  drawn  through  any  point  of 
m>m'  parallel  to  nn',  and  terminated  by  the  perpendicular 
planes,  such  line  will  evidently  be  equal  to  nn',  and  its  ex- 
tremities will  likewise  be  projected  on  the  line  mm',  at  the 
same  points  k  and  I.  Hence,  if  the  property  be  true  for  the 
parallel  line  which  intersects  mm', it  will  likewise  be  true  for 
the  line  nn'. 

369.  It  should  be  observed,  that  in  each  of  these  cases,  the 
projections  will  be  affected  with  contrary  signs,  one  falling 
upon  the  line  mm',  the  other  upon  its  prolongation. 

This  appears  from  an  inspection  of  the  figures  152  and 
155,  and  it  likewise  results  from  the  consideration  that  if  the 
two  projections  fell  upon  the  line  or  upon  its  prolongations, 
the  length  of  nn'  would  necessarily  be  greater  or  less  than 
that  of  mm',  which  by  hypothesis,  is  impossible. 

370.  It  follows  from  the  preceding  remarks,  that  if  we  sup- 
pose two  equal  and  opposite  forces  to  act  in  the  direction  of 
the  line  mm'  on  the  points  m  and  m',  and  denote  by  v  and  v' 
the  projections  of  the  virtual  velocities  mn,  and  m'n'  on  the 
line  of  direction  of  the  forces,  we  shall  have 

V— — v'  ; 
and  consequently,  that  if  we  represent  by  {mm!)  each  of 
these  equal  forces,  we  shall  obtain 

{mm')v  +  {mm')v'=0  ; 
which  proves  that  the  forces  represented  by  {mm')  being 
applied  at  the  extremities  of  the  right  line,  and  being  regarded 
as  sustaining  those  points  in  equilibrio,  the  sum  of  the 
moments  of  the  virtual  velocities  of  these  points  will  be  equal 
to  zero. 

371.  By  the  aid  of  this  proposition  it  will  be  easy  to 

16 


182  STATICS. 

establish  the  principle  of  virtual  velocities  in  the  case  of  any 
number  of  forces  applied  to  different  points.  For,  let  P,  P', 
P",  «fee.  {Fig.  156),  be  several  forces  applied  to  the  points 
m,  in\  m",  <fec.  If  we  regard  these  points  as  firmly  con- 
nected by  inflexible  right  lines,  these  lines  may  be  considered 
as  the  directions  of  equal  and  opposite  forces  acting  on  the 
points  w,  m',  m",  «fee,  and  if  we  denote  these  forces  by  {mm'), 
{7n''ni"),  (fee,  the  equilibrium  will  be  maintained 

at  the  point  m,  by  the  forces  {Tnm/),  {mm")^  {mm'"),  and  P, 
at  the  point  m',  by  the  forces  {m^'m),  {in'ml^,  (m'm'"),  and  P', 
at  the  point  m",  by  the  forces  {m"m),  {tn"m'),  {m"m"')  and  P", 
at  the  point  m'",  by  the  forces  im"'m),  {r)i"'m'),  {m"'7?i"),  andP'", 
&c.  (fee.  (fee. 

Since  the  equilibrium  subsists  at  each  of  these  points,  the 
equation  of  the  moments  obtained  in  Art.  365,  will  manifestly 
be  satisfied.  Let  the  following  notation  then  be  adopted,  viz  ; 
v=projection  of  the  virtual  velocity  of  one  of  the  points  m 
?«',  in",  &c.,  the  point  to  which  this  velocity  refers  being 
designated  by  the  manner  in  which  v  is  written  in  the  ex- 
pression for  its  moment  ;  thus,  v{m?n')  represents  that  v  in 
this  moment  applies  to  the  point  m,  while  v{m'm)  denotes 
that  V  applies  to  7?i'. 

The  character  v  will  thus  represent  quantities  which  may 
be  equal  or  unequal,  according  as  the  projections  of  the 
virtual  velocities  fall  upon  the  same  or  upon  different  lines. 

372.  Having  adopted  this  notation,  the  equations  of  the 
moments  as  given  by  Art.  365,  may  be  expressed  as  follows  : 

for  the  point  m,  P2^-}-v{7nm') -\-v{?7im")  +v{mm"')=0, 
for  the  point  m',  ¥'p' -{-v{7}i'm) +v{7)i'7n")-^v{m'm"')=0, 
for  the  point  7Ji"j  V"p"-\-v{m"7n)-\-v{7?i"77i')-^v{77i"7n'")=0, 
for  the  point  m"',  V"'p'"+v{7n"'m)+v{?Ji"'?7i')+v{??i'"7?i")=0. 

The  sum  of  these  four  equations  being  taken,  we  remark 
that  the  moments  appertaining  to  the  same  right  line  mutu- 
ally destroy  each  other  ;  thus,  the  term  v {771771')  will  cancel 
the  term  v{m'77i),  <fcc.,  and  by  continuing  the  process,  the  sum 
will  be  reduced  to 

rp  -f  P'p'  -i-p>" + P"'7/"=o. 


PRINCIPLE    OF    VIRTUAL    VELOCITIES.  183 

The  same  demonstration  is  evidently  applicable  to  a  greater 
number  of  forces. 

373.  As  an  example  of  the  manner  in  which  the  conditions 
of  equilibrium  in  any  machine  may  be  inferred  from  the 
principle  of  virtual  velocities,  we  will  suppose  the  relation 
between  the  power  and  resistance  in  the  lever  to  be  unknown. 
The  forces  exerted  upon  the  lever  are  the  power  P,  the 
resistance  P',  and  the  reaction  of  the  point  of  support.  If  a 
slight  motion  be  communicated  to  the  lever,  causing  it  to  turn 
about  its  fulcrum,  this  fulcrum  will  remain  immoveable,  and 
the  moment  of  the  reaction  exerted  by  this  point  will  there- 
fore be  equal  to  zero.  Hence,  the  principle  of  virtual  velo- 
cities will  give 

or, 

Pp  =  —Fy (146). 

This  being  premised,  let  the  values  of  the  quantities  jo  andp' 
be  now  determined.  Let  C  represent  the  fulcrum  of  a  lever 
mm'  {Pig.  157),  which  being  slightly  removed  from  its  po- 
sition of  equilibrium  has  assumed  the  position  mi'  ;  the  angles 
at  C  being  equal  to  each  other,  the  arcs  mn,  m'n'  will  be  pro- 
portional to  the  radii  with  which  they  are  described,  and  we 
shall  therefore  have 

mn  :  m'n'  :  :  Cm  :  Cm' (147). 

But  if  through  the  points  n  and  n'  perpendiculars  be  drawn 
to  the  directions  of  the  forces  P  and  P',  we  shall  have 

mr  =  — p ,  m'?-'  —p  ', 
the  negative  sign  being  prefixed  to  p,  because  it  falls  on  the 
prolongation  of  the  force  P.  The  arcs  being  regarded  as  in- 
definitely small  right  lines,  the  right-angled  triangles  mm, 
m'r'n'  will  be  similar  ;  for  the  isosceles  triangles  7nCn,  7n'Cn' 
give 

angle  7imC= angle  n'm'C  : 
and  by  subtracting  these  equal  angles  from  the  right  angles 
rmO,  r'm'C,  there  will  remain 

anffle  7-mn=an2fle  r'm'n'. 
Thus,  the  triangles  rmn,  r'7n'n'  will  be  similar,  and  will  give 
the  proportion 


184  STATICS. 

mn  :  m'n'  ::  mr  :  mY  ; 
or, 

mn  :  m'n'  :  :  — p  :  p'  ; 

and  therefore  the  proportion  (147)  may  be  converted  into 

Cm  :  Cm'  :  :  — p  :  p'. 
But  the  equation  (146)  which  expresses  the  principle  of  vir- 
tual velocities  gives  rise  to  the  proportion 

F  :P::  -p  :p'; 
whence,  by  the  equality  of  ratios, 

Cm  :  Cm'  :  ;  P'  :  P, 
or  the  forces  are  in  the  inverse  ratio  of  the  arms  of  the  lever. 

Of  the  Position  of  the  Centre  of  Gravity  of  a  tSystem  when 
in  Equilibrio. 

374.  Let  m,  m',  m"  (fee,  be  the  centres  of  gravity  of  different 
bodies  which  are  connected  together  in  an  invariable  manner  ; 
let  perpendiculars  z,  z',  z",  &c.,  be  demitted  from  these  points 
on  the  plane  of  xy,  supposed  to  be  horizontal  ;  the  weights 
P,  P',  P",  &.C.,  of  the  several  bodies,  which  may  be  regarded 
as  suspended  from  the  points  m,  m',  m"y  ace,  will  act  along  the 
directions  of  these  perpendiculars.  If  z/  denote  the  co- 
ordinate of  the  centre  of  gravity  of  the  whole  system,  we 
shall  have  (Art.  166) 

P^-fP^^^-fP^s'^+&c. 
^/--     p+P'4-P"+&c.     • 

When  the  system  of  bodies  changes  its  position,  the  ordi- 
nate z  becoming  z  +  hyOr  z — h,  the  increment  of  2;  will  affect 
the  values  of  z',  z'\  z",  (fee,  since  the  points  m,  m\  ?n",  (fcc^ 
being  connected  in  an  invariable  manner,  the  value  of  z. 
cannot  change  without  the  values  of  z',  z",  (fee,  undergoing 
a  corresponding  alteration.  Although  we  are  generally  unac- 
quainted with  the  law  of  dependence  which  exists  between 
the  positions  of  the  different  bodies  composing  the  system, 
the  preceding  equation  may  nevertheless  be  written  under 
the  form 

_Pz-^V'<pz+'P"Fz-\-6cc. 
^'~       P-i-F+P"+(fec.      ' 
in  which  ç,  F,  (fee.  denote  certain  indeterminate  fimctions. 


POSITIONS    OF    THE    CENTRE    OF    GRAVITY.  185 

If  the  value  of  zi  be  a  maximum  or  a  minimum,  the  dif- 
ferential of  the  second  member  will  be  equal  to  zero,  hence 

Vdz + V'd<pz  -f  V'dFz + &c.  =  0  ; 
or, 

Vdz+V'dz'+V"dz"-\-6LC.^Q. 
But  this  equation  is  necessarily  satisfied  when  the  system 
receives  an  infinitely  small  derangement  from  its  position  of 
equilibrium.  For,  when  the  centres  of  gravity  m,  m,  in,  &.C., 
change  their  positions  and  are  transferred  to  ??,  n\  n\  &c., 
the  paths  described  will  be  the  lines  mn,  m'n,  m'n\  &c.  If 
therefore,  these  paths  be  projected  on  the  primitive  directions 
z,  z\  z",  (fee,  of  the  forces  P,  P',  P",  (fee,  we  shall  obtain  the 
values  of  the  projections  of  the  virtual  velocities.  Thus, 
m/i  [Fig.  158)  the  ])rojection  of  mn  upon  the  co-ordinate  z, 
is  equal  to  nk,  the  increment  which  the  value  of  z  has  re- 
ceived in  consequence  of  the  derangement  sustained  by  the 
system  :  the  sign  of  this  increment  may  be  either  positive  or 
negative.  We  shall  therefore  have,  without  reference  to  the 
signs,  p~dz  ;  and  by  applying  the  same  considerations  to  the 
other  co-ordinates,  it  appears  that  the  differçntial  Vdz  + 
Vdz' -\-Vdz"  +  6cQ..,  will  represent  the  same  quantity  as  the 
expression  Pp+P'jo'-}-P"^"-f  (fee.  ;  and  since  the  latter  quan- 
tity becomes  equal  to  zero  when  the  system  is  in  equilibrio, 
according  to  the  principle  of  virtual  velocities,  we  must  like- 
wise have 

Vdz-{-V'dz'-V'P"dz"-\-ôi.c.=Q]  * 

hence,  dz=0,  which  proves  that  the  centre  of  gravity  is  in 
general  situated  at  the  highest  or  lowest  point,  when  the 
system  is  in  a  state  of  equilibrium.  But  this  proposition 
will  not  always  be  true,  since  dz=0  will  not  always  indicate- 
the  existence  of  a  maximum  or  minimum. 

375.  The  converse  of  this  proposition  is  always  true,  viz: 
Jf  the  centre  of  gravity  of  the  system  he  situated  at  the 
highest  or  lowest  point,  the  system  loill  necessarily  he  in  equi- 
librio ;  for,  dz,  will  then  be  equal  to  zero,  and  the  sum  of  the 
moments  of  the  virtual  velocities  will  also  be  equal  to  zero. 


PART    SECOND. 


DYNAMICS. 

OF  THE  LAW  OF  INERTIA. 

376.  Dynamics  has  been  defined  to  be  that  part  of  Me- 
chanics which  treats  of  the  laws  of  motion  of  sohd  bodies. 
We  shall,  in  the  first  place,  establish  as  a  principle  the  general 
law  of  nature,  that  every  body  will  continue  in  the  state  of 
rest  or  motion  in  which  it  may  be  placed,  unless  it  be  acted 
upon  by  some  external  force.  This  indifference  of  matter  to 
a  state  of  motion  or  rest  is  called  inertia.  It  is  a  conse- 
quence of  this  principle  of  inertia  that  one  body  when  struck 
by  another,  exerts  an  effort  of  resistance  to  the  impulsion, 
whilst  acquiring  a  portion  of  the  motion  of  the  striking  body. 
By  this  same  principle,  a  body  having  received  an  impulse, 
must  move  uniformly  in  a  right  line,  if  not  opposed  by  any 
obstacle  :  for  there  can  be  no  reason  why  the  body  should 
deviate  to  one  side  rather  than  to  the  other,  nor  that  its 
motion  should  be  accelerated  rather  than  retarded.  It  is 
true,  that  the  nature  of  the  force  being  unknown  to  us,  we 
cannot  foresee  whether  its  effect  will  be  such  as  to  preserve 
the  motion  of  the  body  invariable  :  thus,  the  law  of  inertia 
should  be  regarded  as  a  simple  result  of  experience  and 
analogy. 

If  we  do  not  perceive  the  motions  of  bodies  to  continue 
unchanged,  it  is  merely  because  these  motions  are  constantly 
affected  by  the  resistance  of  media,  by  the  action  of  gravity, 
or  by  other  similar  causes.  The  most  simple  kind  of  motion 
which  can  be  conceived  is  that  which  takes  place  uniformly, 
and  in  a  right  line. 


188  DYNAMICS. 


Of  Uniform  Rectilinear  Motion. 

377.  A  body  is  said  to  have  a  uniform  motioîi  when  it 
passes  over  equal  spaces  in  successive  equal  portions  of  time  : 
thus,  if  V  denote  the  space  which  it  describes  in  a  unit  of 
time,  it  will  have  described  a  space  2V  at  the  end  of  two 
units  of  time,  3V  at  the  end  of  three  units  of  time,  (fcc.  Con- 
sequently, if  we  represent  by  t  the  number  of  units  of  time 
necessary  for  the  body  to  describe  a  space  s,  this  space  will 
be  equal  to  ^xV  ;  we  shall  thus  have 

s=Yt. 
Such  is  the  equation  of  uniform  motion.  The  coefficient  V, 
or  the  space  passed  over  in  a  unit  of  time,  is  called  the 
velocity,  and  it  evidently  expresses  the  rate  of  a  body's 
motion.  For,  if  a  body  M  move  n  times  as  rapidly  .as 
another  M',  the  space  V  described  by  the  first  in  a  unit  of 
time,  will  obviously  be  n  times  greater  than  the  space  V, 
described  by  the  second  in  the  same  time. 

378.  For  the  purpose  of  comparing  the  circumstances  of 
motion  of  two  bodies  which  depart  at  the  same  instant  from  a 
point  A,  with  velocities  represented  by  V  and  V",  we  will 
denote  by  s'  and  s"  the  respective  spaces  passed  over  by  these 
bodies,  at  the  expiration  of  the  times  i'  and  t"  :  we  shall  then 
have 

whence  we  deduce 

s'  _  Y't'  . 

s"~Y"ï"  ' 
which  proves  that  the  spaces  passed  over  are  proportional  to 
the  products  of  the  times  and  velocities.     When  the  times  are- 
equal,  this  equation  reduces  to 

7'~V"'' 

and  the  spaces  described  are  then  proportional  to  the  ve- 
locities. 

379.  The  body  may  have  alteady  passed  over  a  space  S, 
previous  to  the  instant  from  which  the  time  t  is  reckoned  : 


UNIFORM    RECTILINEAR    MOTION.  189 

we  shall  then  have  the  more  general  equation  of  uniform 
motion 

s=S+Yt, 
in  which  s  represents   the  distance  of  the  body  from  the 
origin  of  spaces.     The  quantity  S  is  called  the  initial  space, 
and  evidently  represents  the  distance  of  the  body  from  the 
origin,  at  the  commencement  of  the  time  t. 

380.  By  the  aid  of  this  equation  we  can  readily  solve  all 
the  problems  of  uniform  rectilinear  motion. 

For  example,  if  the  distance  of  a  body  from  the  origin  of 
spaces  at  the  end  of  the  time  t\  be  supposed  equal  to  s'  ;  and 
if  this  distance  become  s"  at  the  end  of  the  time  t",  we  can 
thence  determine  the  velocity  V,  and  the  initial  space  ;  for 
we  shall  have  the  equations 

s=S+Yt',     s"=S-{-Yf; 
from  which  we  obtain 

~  t"—t'      ~~  t"—t' 

381.  As  a  second  example,  let  it  be  required  to  determine 
the  time  of  meeting  of  two  bodies  M'  and  M  {Pig.  159), 
which  depart  at  the  same  instant  from  the  two  points  A  and  B, 
having  the  respective  velocities  V  and  V.  Let  C  be  their 
point  of  meeting  :  the  spaces  actually  passed  over  by  the  two 
bodies  will  be 

BC=V^,  and  AC=V^ 

If  we  denote  by  h  the  distance  AB  between  the  bodies  at  the 
commencement  of  the  motion,  and  reckon  their  distances  at 
the  end  of  the  time  t  from  the  point  A  as  an  origin^  we  shall 
have  the  equations 

Each  of  the  spaces  5  and  s'  will  then  be  represented  by  the 
line  AC  ;  and  by  placing  the  second  members  of  the  above 
equations  equal  to  each  other,  we  deduce 

_     b 

382.  Since  the  space  &  constantly  varies  with  the  time  /, 


190  DYNAMICS. 

we  can  differentiate  the  equation  s=zS-\-Yt  with  reference  to 
these  two  variables,  and  we  shall  thus  obtain 

dt 
Hence  it  appears,  that  in  uniform  motion  the  velocity  is  the 
differential  coeificient  of  the  space,  regarded  as  a  function 
of  the  time  :  it  will  presently  appear  that  the  same  is  true  in. 
varied  motion. 


Of  Varied  Motion. 

383.  When  the  motion  of  a  body  is  such  that  it  passes  over 
unequal  spaces  in  equal  successive  portions  of  time,  the  body 
is  said  to  have  a  varied  motio?i.  This  kind  of  motion  cannot 
be  produced  by  the  action  of  a  single  force  of  impulsion, 
since  by  the  law  of  inertia  the  velocity  imparted  by  a  single 
impulse  should  constantly  remain  unchanged  ;  and  hence  the 
motion  would  continue  uniform  :  whereas,  we  have  in  the 
present  instance  supposed  it  variable.  It  therefore  becomes 
necessary  to  suppose  that  the  body,  having  received  the  first 
impulsion,  is  subsequently  subjected  to  the  action  of  a  second 
impulse,  a  third,  &-c.,  which,  by  constantly  changing  its 
velocity,  produce  a  variable  motion.  If  the  force  acts  without 
intermission, the  impulses  will  be  communicated  at  intervals 
which  are  indefinitely  small,  and  the  force  is  then  called  an 
incessant  force.  If  the  force  tends  to  increase  the  velocity 
of  the  body,  it  is  called  an  acceleratiîig-  force,  and  when  it 
tends  to  diminish  the  velocity,  a  retarding  force. 

384.  The  velocity  of  the  body  being  supposed  constantly 
variable,  we  can  only  estimate  its  value  at  any  particular 
point  of  the  path  described,  by  supposing  it  to  become  con- 
stant at  this  point.  Thus,  to  measure  the  velocity  of  a  body 
which  has  arrived  at  B  {Fig:  159),  at  the  end  of  the  time  t, 
we  suppose  the  action  of  the  incessant  force  to  be  suddenly 
arrested,  and  the  body  will  then  move  uniformly  with  the 
velocity  which  it  has  acquired  at  the  point  B.  The  space 
BC  described  in  a  unit  of  time,  with  this  uniform  motion,  is 
the  measure  of  the  velocity  at  the  point  B. 


VARIED    MOTION.  191 

385.  The  second  is  usually  adopted  as  the  unit  of  time. 
Hence,  the  velocity  of  the  body  at  the  expiration  of  the  time  t 
will  be  the  space  which  this  body  would  describe  in  the  second 
which  succeeds  the  time  t,  if,  at  the  end  of  the  time  t,  the 
incessant  force  should  cease  to  communicate  new  impulses  to 
the  body. 

386.  To  determine  the  analytical  expression  for  the  ve- 
locity, we  will  suppose  the  body  to  have  arrived  at  the  point 
B,  at  the  expiration  of  the  time  t  ;  the  space  AB  which  it  has 
already  passed  over  being  dependent  on  the  length  of  time 
which  has  elapsed,  the  former  will  evidently  be  a  function 
of  the  latter.  Thus,  we  may  regard  the  space  5  as  the  ordi- 
nate of  a  curve  whose  absciss  is  equal  to  t  ;  consequently, 
when  t  becomes  t  +  dt,  s  will  become  s-\-ds  ;  hence,  the  space 
passed  over  in  the  time  dt  will  be  represented  by  ds.  This 
being  premised,  let  it  be  supposed  that  when  the  body  has 
arrived  at  the  point  B,  the  incessant  force  ceases  to  act  ;  the 
body  will  assume  a  uniform  motion  with  the  velocity  ac- 
quired at  the  point  B,  and  will  describe  in  the  instant  dt 
succeeding  the  time  t,  the  indefinitely  small  space  ds  :  in  the 
next  succeeding  instant  dt  it  will  describe  a  second  space  ds, 
and  the  same  will  continue  until  the  body  has  described  a 
space  BC,  which  will  correspond  to  the  unit  of  time.  This 
space  BC  will  therefore  contain  ds  as  many  times  as  dt  is 

contained  in  unity  ;  but  —  will  express  the  number  of  times 

which  the  unit  of  time  contains  the  quantity  dt  ;  hence,  the 

1  ds 

space  BC  will  be  expressed  by  ds  X  -^,  or  by  —,  since  the  dif- 

(it  (it 

ferential  is  taken  with  reference  to  the  variable  t  ;  but  the 
space  BC  represents  the  quantity  v  ;  we  shall  therefore  have, 
for  the  expression  of  the  velocity  in  varied  motion 

ds 

dt 

387.  It  may  also  be  observed  that  the  space  passed  over, 
after  the  expiration  of  the  time  t,  will  be  {Fig.  159), 

B6=c?5  at  the  end  of  the  time  dt, 
Bb'=2ds  at  the  end  of  the  time  2dt, 


192  DYNAMICS. 

Bb"=3ds  at  the  end  of  the  time  3dt. 

BC=iids  at  the  end  of  the  time  ?i .  dt. 
And  since  the  time  elapsed  during  the  passage  of  the  body 
from  B  to  C  is  by  hypothesis  equal  to  unity,  we  may  suppose 

7uit~l\   whence  n=--.     This   value   beinff  substituted  in 
dt 

the  expression  nds=v,  the  space  described  in  a  unit  of  time 

we  shall  obtain,  as  above, 

"4:  •••■("«> 

388.  Before  investigating  the  expression  for  the  value  of 
the  incessant  force,  it  will  be  necessary  to  discover  the  rela- 
tion which  exists  between  the  force  and  the  velocity. 

If  a  force  P  be  supposed  to  communicate  a  velocity  v  to 
any  body,  a  force  7i  times  as  great  will  communicate  to  the 
body  a  velocity  equal  to  nv.  The  truth  of  this  proposition 
might  well  be  questioned,  since  the  nature  of  forces  being 
entirely  unknown,  we  cannot  affirm  that  a  double  force  will 
necessarily  produce  a  double  velocity  ;  or,  in  general,  that  a 
single  force  equal  to  the  sum  of  two  others,  will  necessarily 
produce  a  velocity  equal  to  the  sum  of  the  velocities  which 
the  two  forces  would  separately  produce.  But  the  fact  being 
confirmed  by  universal  experience,  we  adopt  it  as  a  principle. 
Thus,  by  supposing  different  forces  applied  to  the  same  body 
or  material  point,  their  relative  intensities  can  be  estimated 
by  comparing  the  velocities  which  they  would  severally  com- 
municate. 

The  proper  measure  of  an  incessant  force  will  be  the 
velocity  which  it  can  generate  in  a  given  time  ;  but  the  in- 
tensity of  the  force  being  constantly  variable,  we  must  sup- 
pose the  force  to  become  constant  at  the  instant  when  we 
wish  to  estimate  its  value,  and  the  measure  of  the  force  will 
then  be  the  velocity  generated  in  the  unit  of  time  succeeding 
this  instant.  The  velocity  communicated  by  this  incessant 
force  during  the  unit  of  time,  when  it  is  supposed  to  retain  a 
constant  value,  will  obviously  be  unequal  to  that  which 
would  have  been  communicated  b]f  the  variable  incessant 
force,  in  the  same  time. 

389.  The  preceding  remarks  indicate  the  method  of  meas- 


VARIED    MOTION.  198 

uring  the  incessa,nt  force  ;  since  they  determine  the  ratio  in 
which  the  intensity  of  the  force  varies  in  different  times. 

If,  for  example,  at  the  expiration  of  the  times  t  and  t\  the 
incessant  force,  having  become  constant,  can  generate  in  a 
second  of  time  velocities  represented  by  the  numbers  60  and 
20,  we  infer  that  the  intensity  of  the  force  at  the  end  of  the 
time  t  is  triple  its  intensity  at  the  end  of  the  time  t' . 

390.  To  deduce  from  the  above  definition  the  analytical 
expression  for  the  incessant  force,  let  v  represent  the  velocity 
acquired  by  the  body  at  the  end  of  the  time  t  ;  then,  at  the 
expiration  of  the  time  t-\-dt^  the  velocity  will  become  v^dv  ; 
consequently,  dv  will  be  the  velocity  communicated  during 
the  time  dt  ;  but  if  at  the  end  of  the  time  t  the  intensity  of 
the  force  be  supposed  to  become  constant,  there  will  be  com- 
municated to  the  body  in  the  instant  dt  which  succeeds  the 
time  /,  a  velocity  represented  by  dv  ;  and  the  same  effect  will 
be  repeated  during  any  number  of  succeeding  instants  ;  so 
that  the  velocities  communicated  after  the  expiration  of  the 
time  t^  in  the  instants  dt,2dt,  3dt,  &c.,  will  be  expressed  by  dv, 
2dv,  3dv,  &.C.  :  and  consequently,  the  velocity  communicated 
in  the  unit  of  time  which  succeeds  the  time  t,  will  be  equal  to 
dv  repeated  as  many  times  as  dt  is  contained  in  unity.     This 

number  being  expressed  by  -— ,  it  follows  that  —  x  dv,  or  — , 

dt  dt  dt 

will  express  the  effect  of  the  force  or  the  velocity  generated  in 

a  unit  of  time.     If,  therefore,  we  denote  this  force  by  ^,we 

shall  obtain  for  the  second  equation  of  varied  motion, 

^4 ("«)• 

The  character  ç  will  hereafter  be  used  to  designate  the  inten- 
sity of  the  force  ;  the  force  being  represented  by  the  effect 
which  it  produces. 

391.  From  the  preceding  equation  we  obtain 

çdt—dv] 
thus,  if  the  incessant  force  be  given,  the  increment  to  the 
velocity  in  the  time  dt  can  be  readily  calculated. 

392.  By  eliminating  dt  between  the  equations  (148)  and 
(1 49),  we  obtain  a  third  equation  of  varied  motion, 

çds=vdv. 

N  17 


194 


DYNAMICS. 


Of  Uniformly  Varied  Motion. 

393.  The  incessant  force  imparting  at  each  instant  a  new 
impulse  to  the  body,  if  these  impulses  are  equal  in  intensity, 
the  body  will  acquire  the  same  velocity  in  a  unit  of  time  after 
the  expiration  of  the  time  t,  as  it  would  after  a  time  t'.  Let 
this  velocity  which  is  constantly  generated  in  a  unit  of  time, 
be  denoted  by  ^  ;  we  shall  then  have 

Substituting  this  value  in  the  equation 

^     dv 
'^=dt^ 
we  shall  obtain 

dv=gdt] 
and  by  integrating  and  denoting  by  a  the  constant  which  will 
thus  be  introduced,  we  find 

v=a+gt (150).* 

We  have  likewise  obtained  for'the  value  of  the  velocity 

ds 

hence,  if  we  eliminate  v  between  these  two  equations,  we 
shall  have 

ds={a-i-gt)dt, 
from  which,  by  integration,  we  find 

s=b  +  at+^gt' (151), 

the  quantity  b  being  an  arbitrary  constant. 

*  This  equation  might  also  have  been  obtained  from  the  following  considera- 
tions :  Let  it  be  supposed  that  a  body  in  motion  has  acquired  a  velocity  a:  if  it 
then  be  solicited  by  a  constant  force  which  communicates  to  it  a  velocity  g  in 
each  second  of  time,  the  velocity  of  the  body  will  become 

a-\-g,  at  the  end  of  one  second, 

a-\-2g,  at  the  end  of  two  seconds, 

a-{-3g,  at  the  end  of  three  seconds, 


a-\-tg,  at  the  end  of  t  seconda  : 
thus,  if  we  represent  by  v  the  velocity  of  the  body  at  the  expiration  of  the  time  /, 
we  shall  have 


UNIFORMLY    VARIED    MOTION.  195 

If  ^  be  supposed  positive  in  this  equation,  the  motion  will 
be  uniformly  accelerated,  but  if  negative,  the  motion  will  be 
uniformly  retarded. 

394.  If  we  make  ^=0,  we  find  h=s\  thus,  h  will  represent 
the  initial  space,  or  the  distance  of  the  body  from  the  origin, 
at  the  instant  from  which  the  time  is  reckoned. 

The  constant  a  is  equal  to  the  initial  velocity  of  the  body, 
as  appears  by  making  ^=0  in  equation  (150). 

395.  When  the  initial  space  and  initial  velocity  are  each 
equal  to  zero,  the  equations  (150)  and  (151)  become 

•^^gt (152), 

s^\gt^ (153), 

and  the  body  then  moves  from  rest,  under  the  action  of  the 
incessant  force. 

396.  Let  s  and  s'  represent  the  spaces  described  in  the 
times  t  and  t',  under  the  action  of  a  force  g  ;  the  equation 
(153)  gives 

s=^\gf,    and  s'==\gV* (154)  ; 

whence  we  obtain  the  proportion 

s:s'  \:t^  '.t'^ (155). 

Consequently,  the  spaces  described  by  a  body  in  different 
times,  when  it  moves  from  rest,  being  solicited  by  a  constant 
accelerating  force,  are  proportional  to  the  squares  of  those 
times. 

397.  The  equation  (152)  gives 

v=gt,     and  v'=gt', 
whence, 

V  :v'  ::t:  t', 
and  by  comparing  this  proportion  with  (155),  we  have 

v:v'::^s:  ^s'. 
Hence  it  appears  that  the  tim,es  elapsed  are  constantly  pro- 
portional to  the  velocities,  or  to  the  square  roots  of  the  spaces 
described  in  those  times. 

398.  If  we  make  ^=1,  the  equation  (153)  becomes 

sz=ig. 

In  this  case,  s  represents  the  space  described  by  the  body 
in  the  first  unit  of  time,  and  it  appears  that  this  space  is 

N2 


1^  DYNAMICS. 

equal  to  one-half  the  quantity  g,  which  represents  the  mea- 
sure of  the  accelerating  force.  It  has  been  found,  for  example) 
that  a  body  subjected  to  the  action  of  gravity,  would  describe 
in  the  first  second  of  time,  in  the  latitude  of  New-York  a 
distance  equal  to 

16.0799  feet,  or  nearly  16j^  feet  ; 
this  value  being  substituted  in  tht?  place  of  s  in  the  preceding 
equation,  we  find 

^=32.1598  feet,  or  nearly  =  32i  feet. 

399.  The  equation  (153)  will  determine  the  space  described 
in  a  given  time  ;  for  example,  if  t=6",  we  shall  have 

5  =  1^^2^1(321")  x36=579  feet; 

thus  a  body  being  elevated  to  the  height  of  579  feet,  would 
require  six  seconds  to  fall  to  the  surface  of  the  earth. 

400.  The  velocity  acquired  by  this  body,  when  it  has  reached 
the  surface,  may  be  determined  from  equation  (150),  in  which 
we  make 

a=0,    g=32i  feet,     ^=6". 

We  thus  find 

v=32i"x  6=193"-. 

401.  If  it  be  required  to  determine  the  height  from  which 
a  body  must  fall  to  acquire  a  given  velocity,  we  eliminate  ( 
between  the  equations 

and  we  thus  obtain 

^=v/(2^*) (156). 

Let  it  be  supposed,  for  example,  that  we  wish  to  détermine 
the  space  through  which  a  body  would  fall  in  acquiring  a 
velocity  of  386  feet  per  second  ;  we  shall  have 

386"=.,/(2 X 321"- X5)=</(64i"- X5)  ; 

whence, 

(386")^_148996;;;_ 

The  velocity  acquired  in  falling  through  a  given  height  is 
called  the  velocity  due  to  that  height. 

402.  To  determine  tlie  time  in  which   a  body  will  fall 


UPWARD    VERTICAL    MOTION.  197 

through  a  given  height  s,  we  employ  the  equation  (153), 
which  gives 


'=v/(|) 


403.  The  general  equations  of  variable  motion 

ds  dv  ,1  -^. 

ir"'  *=* (1'^' 

will  now  be  applied  to  the  investigation  of  the  circumstances 
of  varied  motion  under  different  hypotheses.  This  investi- 
gation is  reduced  to  the  determination  of  the  relations  which 
exist  between  the  time  elapsed,  the  space  described,  and  the 
velocity  acquired,  since,  if  the  two  latter  can  be  expressed  in 
functions  of  the  time,  we  shall  be  able  to  discover  the  place 
of  the  body,  and  the  velocity  with  which  it  moves  at  any 
given  instant.  Thus,  the  circumstances  of  motion  will  be 
entirely  known. 

Of  the  Motion  of  a  Body  projected  Vertically  upward. 

404.  When  the  action  of  gravity  is  alone  exerted  on  a  body^ 
we  have  the  relation 

v=gt, 

in  which  v  expresses  the  velocity  at  the  end  of  the  time  t  : 

but  if  we  suppose  the  body  instead  of  moving  from  rest,  to 

be  projected  vertically  in  a  direction  opposed  to  that  of  gravity^ 

with  a  velocity  «,  this  velocity  will  have  been  diminished  at 

the  end  of  the  time  t,  by  a  quantity  equal  to  the  velocity 

which  gravity  could  impart  in  the  same  time  ;  consequently, 

the  velocity  of  the  body  at  the  expiration  of  the  time  t  will 

be  represented  by  a—gt]  and  if  we  represent  this  velocity 

by  V,  we  shall  have 

v=a—gt (158)  : 

ds 
substituting  for  v,  its  value  -^,  we  find,  by  integration, 

s=at—\gtK 
The  initial  space  being  supposed  equal  to  zero,  no  constant 
has  been  added  in  this  integration. 


198  DYNAMICS. 

This  equation  being  placed  under  the  form 

if  we  substitute  for  t  its  value  deduced  from  equation  (158), 
we  shall  obtain 

a  +  v     a — V 


2  ^-    ' 

or, 

405.  The  equations  (158)  and  (159)  make  known  all  the 
circumstances  of  the  motion  under  consideration.  Thus,, 
the  equî^tion  (158)  indicates  that  the  velocity  constantly  de- 
creases as  the  time  increases  ;  and  the  equation  (159)  proves 
that  the  velocity  decreases  as  the  space  described  becomes 
greater  :  hence,  the  velocity  constantly  becomes  less  as  the 
body  rises.  When  this  velocity  becomes  equal  to  zero,  the 
body  has  attained  its  greatest  elevation  :  if  we  denote  this 
elevation  by  h,  the  equation  (159)  will  give,  by  making  v=0, 

■  *=J (160). 

To  determine  the  time  corresponding  to  this  elevation,  we 
make  v=0,  in  equation  (158),  and  thence  deduce 

t=- (161). 

§■ 
The  velocity  due  to  the  height  h  is  found  by  making  h—» 
in  the  formula 

and  by  substituting  the  value  of  h  deduced  from  equation 
(160),  we  obtam 

hence,  the  body  acquires  the  same  velocity  in  descending,  that 
it  lost  in  ascending. 

406.  Let  it  be  required  to  determine  the  greatest  height  to. 
which  a  body  will  rise  when  projected  vertically  upward  with 
a  velocity  of  100  feet  per  second  :  we  shall  find  from  equa- 
tions (160)  and  (161),  that  the  greatest  height  is  155 -^Vô  feet. 


VERTICAL    MOTION   OP   A   BODY.  ISS' 

and  that  the  time  of  rising  or  falling  is  equal  to  3i  seconds, 
nearly. 

407.  The  preceding  equations  may  likewise  be  applied  to 
the  case  in  which  the  body  is  projected  downward,  by  simply 
changing  the  sign  of  the  quantity  g  ;  we  shall  thus  have  an 
expression  for  the  velocity  f , 

v=a-{-gt. 

Of  the  Vertical  Motion^of  a  Body  when  acted  upon  by  the 
Force  of  Gravity  considered  as  variable. 

408.  Gravity  is  a  force  whose  intensity  varies  at  different 
distances  from  the  earth's  centre.  The  law  of  this  variation 
has  been  discovered  to  be  that  of  the  inverse  ratio  of  the 
square  of  the  distance  ;  that  is  to  say,  that  at  distances  from 
the  centre  of  the  earth  represented  by  2,  3,  4,  dec,  it  becomes 

•—,  --,  —,  &c.,  of  its  value  at  the  distance  unity.      Thus, 

/i         Ô        4' 

although  a  body  falls  through  a  distance  of  1^-^^  feet  in  the 
first  second  of  time  at  the  surface  of  the  earth,  it  would  fall 
through  a  much  less  space  in  the  same  time  if  the  distance 
of  the  body  from  the  centre  were  greatly  increased. 

409.  Let  a  body  be  supposed  to  depart  from  rest  at  the 
point  A  {Pig-  160),  and  let  it  be  required  to  ascertain  the 
velocity  of  the  body  when  it  has  reached  the  point  B.  Denote 
by  g  the  intensity  of  the  force  of  gravity  at  M,  the  surface  of 
the  earth,  and  by  ç>  its  intensity  at  the  point  B  ;  by  r  the 
radius  of  the  earth  CM,  and  by  x  the  distance  from  B  to  C  : 
for  the  purpose  of  simplifying  the  calculation,  let  the  known 
distance  AC  be  assumed  as  the  linear  unit.  The  force  being 
supposed  to  vary  in  the  inverse  ratio  of  the  square  of  the 
distance  from  the  earth's  centre,  we  shall  have 

g  i<p  :ix^  :  r^  ; 
whence, 

But  the  general  expression  for  the  incessant  force  being 

dv 


200  DYNAMICS. 

we  shall  obtain,  by  placing  these  values  of  ^  equal  to  each  other, 
^=^-11- (162). 

Again,  the  velocity  being  equal  to  the  differential  of  the  space 
divided  by  the  differential  of  the  time,  it  will  be  represented  by 

dt   ' 

or  by  its  equal 

"=-'7 (i«^)- 

Multiplying  the  terms  of  this  equation  by  the  corresponding 
terms  of  equation  (162),  we  find 


and  by  integration, 


,  „dx 

vdv= — ^r^ — , 


The  constant  may  be  determined  from  the  consideration  that 
when  a:=AC=l,  i;=0;  hence. 

This  value  substituted  in  the  preceding  equation  gives 


j=^'-il~') (^«^)- 


This  equation  determines  the  value  of  the  velocity  at  any 
given  point  of  the  line  AC. 

410.  To  determine  the  time  employed  by  the  body  in 
describing  the  space  AB,  we  eliminate  v,  between  this  equa- 
tion and  the  equation  (163),  and  we  thus  obtain 

2di^-^''  \x     V' 
whence, 

dt'=- ^x 


2^r^''l_^' 

X 

and  consequently, 

dx 


"-v^va-o' 


VERTICAL    MOTION    OF    A    BODY.  201 

by  the  integration  of  this  equation  we  shall  obtain 

--^4/^1^ (-)■ 

To  effect  the  integration  which  is  here  only  indicated,  we 
reduce  the  fraction  to  a  simpler  form,  thus 

The  radical  in  the  denominator  may  be  caused  to  disappear, 
by  making 

l—x=z'. 
We  deduce  from  this  equation, 

These  values  substituted  in  the  preceding  formula  give 


ng  by  parts,  we  find 


•(1-4 

Integrating  by  parts,  we  find 

z'^dz 

But  we  likewise  have  the  identical  equation 

rj       /-.         .\      P     dz  /*    z^dz 

Adding  these  equations,  and  dividing  by  2,  we  obtain 

fdz^{i-z^)==^,z^{i-zn+^f^^;^^^) 

=^z^{l—z'^}  +  \  arc  (sin=z)  ; 
consequently, 

—2fdz^{l—z')  =  —z^{l—z'^)—a.rc{smz=z), 
and  by  substituting  this  value  in  the  equation  (167),  we  shall 
obtain  the  integral  of  (166)  ;  hence,  the  equation  (165)  will 
become 

i=±~./hz^(l-z')+  arc  (sin=2:)] (168). 

The  constant  will  be  equal  to  zero,  since  a-=:1,  when  ^=0  ; 
and  therefore,  2;  =  y/(l—.r)=0;  this  supposition  causes  the 


202  DYNAMICS. 

second  member  of  the  equation  to  vanish.  Moreover,  the 
time  being  essentially  positive,  we  use  only  the  inferior  sign 
in  the  preceding  equation  ;  and  by  observing  that  z'  =i  — x= 
the  distance  AB  which  the  body  has  described,  we  shall  have, 
by  representing  this  distance  by  s, 

^=-\/2i-'^x/V(l-*)+arc  (sin=^5)]. 

411.  This  last  equation  is  much  simplified  by  supposing 
the  distances  AB  and  AM  to  be  exceedingly  small  when  com- 
pared with  the  distances  AC  and  MC  ;  for,  the  quantity 
^(1 — s)  may  then,  without  sensible  error,  be  supposed  equal 
to  unity;  and  the  arc  (sin  =  y'5)  may  likewise  be  considered 
as  equal  to  its  sine  ;  hence,  by  changing  r  into  unity,  the 
preceding  expression  will  reduce  to 

and  from  this  we  deduce  the  relation 

or,  the  motion  is  then  similar  to  that  which  would  take  place 
if  the  intensity  of  the  force  remained  invariable. 

Of  the  Vertical  Motion  of  a  Body  in  a  resisting  Medium. 

412.  It  has  been  ascertained  that  a  body  when  moving  in 
a  fluid  experiences  a  resistance  which  is  proportional  to  the 
square  of  the  velocity.  Thus,  by  calling  w  the  intensity  of 
this  resistance  when  the  velocity  of  the  body  is  represented 
by  unity,  the  resistance  will  be  expressed  by  mv^,  when  the 
body  has  acquired  a  velocity  v. 

413.  This  force  being  opposed  to  that  of  gravity  when  the 
body  descends,  we  shall  have,  by  supposing  the  intensity  of 
gravity  constant, 

<p=g—mv'  ; 

dv 
and  by  substituting  for  ç  its  general  value  —,  we  obtain 

dv 


VERTICAL    MOTION   OF    A   BODY.  203 

whence, 

dt=-    ^"^       (169). 

g — mv' 

To  integrate  this  equation,  we  decompose  the  denominator 
into  factors,  and  thus  have 

If  we  then  suppose,  according  to  the  method  of  rational  frac- 
tions, 

^"      =^dv( ^— +  — ? \ (170), 

g—mv^  \^g  +  v^m     y/g—v^m; 

we  shall  find,  by  reducing  the  terms  of  the  second  member 
to  a  common  denominator,  and  placing  the  coeiRcients  of 
the  like  powers  of  v  equal  to  each  other, 

A=B=-— ; 

these  values  substituted  in  the  equation  (170),  give 
dv  1     /        dv  dv         \ 

g — mv''  ~~  2v^  V  ^g  -\-v^m     ^g — ■y-/m  /  * 

Multiplying  and  dividing  the  second  member  of  this  equation 
by  y/m^  we  shall  obtain  a  value,  which  substituted  in  equa- 
tion (169)  will  reduce  it  to 

.  1         /     d'Oy/m  dv^m     \ 

~ 2»/7n^g\^g-^v^m      ^g—v^m)  ' 
and  by  integration,  we  obtain 

^^2^(mg)i^^^  (v^^+^^^Z»^)  -log(v/g--i'v^m))  -f  C  ; 
or, 

t=  -J— loff  ^/S+'^^/m  ,y^^. 

^\/{^g)        \^g-vx/m ^       ^' 

The  constant  may  be  suppressed,  since  when  ^=0,  ^=0. 
414.  If  the  two  members  of  the  equation  (171)  be  multi- 
plied by  2^mg,  and  the  first  member  by  the  logarithm  of 
the  base  e  of  the  Naperian  system,  which  is  equal  to  unity, 
we  shall  have 

2V(^ff).logg=log.^^+^^^, 


204  DYNAMICS. 

or, 

'/g—Vy/m 
and  by  passing  to  the  numbers,  we  have 

415.  This  equation  being  written  under  the  form 

. 'i-__x/g—Vx/ni  ,^^2^ 

e^Vc^e)     ^g+v^7n 

it  is  obvious  that  if  t  be  supposed  to  increase  indefinitely,  the 
value  of  the  first  member  will  approach  to  zero  ;  and  conse- 
quently, when  t  becomes  infinite,  we  shall  have 

^g—Vé/m^O (173) 

From  this  equation  we  deduce 

v—^!-^=di  constant  quantity. 

Hence  we  conclude,  that  as  the  time  increases  the  velocity 
becomes  more  nearly  constant. 

416.  To  determine  the  space  described  in  functions  of  the 
velocity,  we  multiply  the  corresponding  terms  of  the  equations 


dv  ,  ds 

and  we  thus  find 

vdv = {g — mv'^  )ds  ; 

whence, 

J  vdv  .^^.. 

ds= (174). 

g — niv^  ' 

This  equation  may  be  rendered  integrable  by  making 

g — mv^  —z. 

For,  we  obtain  by  differentiation, 

,  dz 

and  these  values  substituted  in  equation  (174),  transform  it 
into 


MOTION    UPON    AN    INCLINED    PLANE.  205 

the  integral  of  which  is 

or,  by  replacing  z  by  its  vakie  g — mv^,  we  have 

*=-2^1og(^-mt72)+C. 

The  constant  C  may  be  determined  by  making  5=0,  and 
t;=0;  whence, 

C=ilog^; 

which  value  being  substituted   in  the  preceding  equation, 
gives 

"»=  2^[log  g—^og  (g-mv')]  ; 
or,  finally, 

'5=-7r-  log  ( ^^-vl- 

2m  \g—mv^  / 

Of  the  Motions  of  Bodies  upon  Inclined  Planes. 

417.  Let  a  body  be  situated  upon  an  inclined  plane,  and 
let  the  weight  of  this  body,  considered  as  a  vertical  force  ap- 
plied at  its  centre  of  gravity,  be  resolved  into  two  components, 
which  shall  be  respectively  parallel  a  id  perpendicular  to  the 
surface  of  the  plane.  The  perpendicular  force,  being  sup- 
posed to  pass  through  a  point  of  contact,  will  evidently  be 
destroyed  by  the  resistance  of  the  plane,  while  the  parallel 
component  will  cause  the  centre  of  gravity  to  describe  a  line 
parallel  to  the  plane.  The  question  will  thus  be  reduced  to 
the  consideration  of  the  motion  of  a  material  point  upon  the 
inclined  plane. 

418.  Let  m  represent  the  material  point  {Pig.  161),  and  g 
the  velocity  which  gravity  can  impart  in  a  unit  of  time  :  if 
the  force  of  gravity,  represented  by  the  vertical  line  wiB,  be 
resolved  into  two  components  nîD  and  mC,  respectively  parallel 
and  perpendicular  to  the  plane,  the  latter  will  be  destroyed  by 
the  resistance  of  the  plane,  and  the  former  will  cause  the 
material  point  to  slide  along  the  plane. 

18 


206 


DYNAMICS. 


But,  since  forces  are  proportional  to  the  velocities  which 
they  communicate  in  the  same  time,  if  we  denote  by  g'  the 
velocity  communicated  in  a  unit  of  time  by  the  component 
which  acts  in  the  direction  of  the  plane,  we  shall  have 

mB  :  7nD  '.:  g  :  g'. 
The  ratio  between  mB  and  niD  being  the  same  as  that  between 
the  length  and  the  height  of  the  plane,  we  shall   have,  by 
representing  these  quantities  by  It'  and  ]i  respectively, 

g:g'::h':h] 
whence, 

^'=f (175). 

419.  From  this  equation  it  appears,  that  the  velocity  g', 
which  is  generated  in  a  unit  of  time  by  the  component  of  gra- 
vity parallel  to  the  plane,  is  equal  to  the  velocity^,  multiplied 

by  the  constant  ratio  —  ;  and  we  therefore  conclude  that  the 

force  which  urges  the  body  along  the  inclined  plane  differs 
from  the  force  of  gravity  only  in  its  intensity.  Hence,  if  we 
denote  by  t'  the  time  requisite  to  describe  the  entire  distance 
mA=h',  the  same  relations  will  exist  between  the  quantities 
g\  h',  and  t',  as  have  been  already  obtained  between  g,h,  and 
t,  in  investigating  the  circumstances  of  uniformly  varied 
motion  :  we  shall  therefore  have 

h'=^ig'r- (176)  ; 

and  the  velocity  acquired  by  the  body  at  the  point  A  will  be 

v'=g't'; 
or  by  eliminating  the  time  i',  we  shall  find 

v'=^{2g'h'). 
If  in  this  equation  we  substitute  for  g'  its  value  found  in 
equation  (175),  we  shall  obtain,  after  reduction, 
v'=^{2gh). 

The  expression  for  the  velocity  being  independent  of  the 
angle  /«AE,  which  the  inclined  plane  forms  with  the  horizon, 
it  follows  that  if  several  bodies  be  allowed  to  descend  from 
the  same  point  m  upon  different  inclined  planes  ?nA,  ?nA', 
mA",  <Scc.  (Mg.  162),  they  will  all  have  ac(]^uired  the  same 


MOTION  UPON  AN  INCLINED  PLANE.        207 

velocity  when  they  shall  have  arrived  at  the  same  horizontal 
plane. 

420.  Although  the  velocities  acquired  at  the  points  A  and 
E  are  equal,  the  times  of  descent  will  be  unequal  ;  for,  if  t 
and  i'  represent  the  times  of  describing  mE  and  mA,  their 
values  will  result  from  the  equations 

'2h       ..  /2h' 


▼  o-  ~         rr' 


but  we  have 


g>g',    or-<--; 
ff     g 

and  from  these  inequalities  we  deduce 

2h    2h' 

7    F' 
which  proves  that  the  value  of  t'  exceeds  that  of  t. 

421,  In  general,  if  t'  and  t"  represent  the  times  of  describ- 
ing two  inclined  planes  A' and  h",  having  a  common  altitude 
A  ;  and  if  g'  and  g-"  represent  the  components  of  gravity 
respectively  parallel  to  these  planes,  we  shall  have 

whence, 

or  by  replacing  g'  and  g"  by  their  values  (Art.  418),  we 
obtain 

Thus,  the  times  of  describing  different  inclined  planes  hav- 
ing a  common  altitude  will  be  proportional  to  the  lengths  of 
those  planes. 

422.  The  motions  of  bodies  upon  inclined  planes  give  rise 
to  a  remarkable  mechanical  property  of  the  circle  :  it  consists 
in  this,— that  if  the  plane  of  the  circle  be  supposed  vertical, 
the  body  will  require  the  same  time  to  describe  a  chord  AC 
(Fig.  163),  as  is  necessary  to  fall  through  the  vertical  diam- 


208 


DYNAMICS. 


eter  AB.     For,  the  equation  (176)  gives,   for   the  time  of 
descent  through  AC, 

^     g 
and  by  substituting  for  g-'  its  vaUie  ^ ,  this  equation  will  be- 


come 


t'=x/^ (177). 


gh 

But  if  the  diameter  of  the  circle  be  denoted  by  d,  we  shall 
have,  by  the  property  of  the  circle, 

AB  :  AC  :  :  AC  :  AD  ; 

or, 

d:h'::  h'  :  h  ; 

and  consequently, 

This  value  substituted  in  equation  (177),  gives,  after  reduction, 

^        g 

but  this  value  is  precisely  the  same  as  that  which  has  been 
found  for  the  time  t,  in  which  the  body  would  fall  through 
the  diameter  AB  :  for,  the  height  AB  being  expressed  by  d, 
we  shall  have 

whence, 

/2d 


Of  Curvilinear  Motion. 

423.  We  have  hitherto  supposed  the  motion  under  consid- 
eration to  be  rectilinear  ;  but  if  it  be  curvilinear,  the  space 
described,  and  the  velocity  acquired  in  a  given  time,  will  be 
insufficient  to  determine  all  the  circumstances  of  the  motion: 
it  will  likewise  be  necessary  to  know  the  nature  of  the  curve 
described  by  the  body,  and  tlie  point  of  this  curve  at  which 
the  body  is  found  at  the  end  of  a  given  time. 


CURVILINEAR    MOTION.  209 

424.  In  the  resolution  of  this  problem,  we  employ  the  prin- 
ciple of  the  parallelogram  of  velocities,  which  is  similar  to 
that  of  the  parallelogram  of  forces.  It  may  be  enimciated  as 
follows  :  If  two  forces  P  and  Q,  {Fig.  164)  communicate^  in 
a  imit  of  time,  to  a  m,aterial  point  m,  velocities  represented 
by  wB  a?id  mC  respectively,  the  resultant  R  q/"  P  and  Q,  will 
communicate  to  the  point,  in  the  same  time,  a  velocity  7wD, 
which  will  he  represented  hy  the  diagonal  of  the  parallelo- 
gram constructed  on  the  lines  mB  and  mC.  The  truth  of 
this  proposition  may  be  thus  established  : — Let  the  force  P  be 
represented  by  the  line  ?nB  ;  then,  since  forces  are  proportional 
to  the  velocities  which  they  communicate  in  a  given  time, 
the  force  Gi  will  be  represented  by  the  line  mC.  But,  by 
regarding  mBDC  as  the  parallelogram  of  forces,  the  diagonal 
mD  will  represent  the  resultant  of  the  forces  P  and  Q,  ;  and 
it  is  required  to  prove  that  the  velocity  resulting  from  the 
composition  of  the  two  velocities  wiB  and  tnC  is  the  same  as 
that  which  is  due  to  the  force  R.  Let  a;  represent  the  velocity 
which  the  force  R  can  communicate  to  the  point  m  in  a  unit 
of  time  ;  then,  since  forces  are  proportional  to  the  velocities 
which  they  generate,  we  shall  have 

V  :R::mB:x. 
But  from  the  parallelogram  of  forces,  we  deduce 

P  :  R  :  :  »iB  :  mD  ; 
hence, 

mB  :  niD  :  :  mB  :  x  ; 
and  therefore, 

x=mD. 

425.  In  the  preceding  remarks  the  forces  P,  Q.,  and  R  have 
been  supposed  to  act  incessantly,  communicating  new  im- 
pulses at  each  successive  instant  of  time.  The  results 
obtained  will  however  be  equally  true  if  we  regard  P,  Q,, 
and  R  as  impulsive  forces  which  communicate  their  effects  in- 
stantaneously, since  the  velocities  imparted  by  such  forces 
are  proportional  to  the  intensities  of  the  forces. 

426.  The  composition  of  three  velocities  by  the  construc- 
tion of  a  parallelepiped,  results  immediately  from  the  pre- 
ceding principle  ;  for,  let  P,  Q,  and  R  {Pig.  165)  represent 

O 


210  DYNAMICS, 

-three  forces  which  communicate  the  velocities  mp^  mq,  and 
mr  to  the  material  point  m  ;  let  the  velocities  mp  and  mq  be 
compounded  into  a  sino;le  velocity  mp\  which,  by  the  pre- 
ceding demonstration,  will  be  the  same  as  that  communicated 
by  the  force  P',  the  resultant  of  the  two  forces  P  and  Q,  :  in 
like  manner,  the  resultant  ms  of  the  two  velocities  mp'  and  mr, 
will  represent  the  velocity  communicated  by  the  force  S,  the 
resultant  of  the  two  forces  P'  and  R,  or  of  the  three  forces 
P,  Q,  and  R  ;  hence,  the  diagonal  of  the  parallelepiped  con- 
structed on  the  lines  representing  the  three  velocities  will 
represent  the  velocity  communicated  by  the  resultant  of  the 
three  forces  P,  Q,,  and  R. 

427.  We  will  now  examine  the  circumstances  in  which  a 
material  point  will  describe  a  curvilinear  path.  For  this 
purpose,  let  the  material  point  m  {Pig.  166),  at  rest,  be  sup- 
posed to  yield  to  the  effect  of  an  impulsion  which  causes  it 
to  describe  the  right  line  mK  in  the  time  6,  and  at  the  end  of 
this  time  let  it  receive  a  iiecond  impulsion  capable  of  making 
it  describe  the  line  AB  in  the  same  time  6  ;  the  material  point 
will  not  entirely  yield  to  the  action  of  this  second  force,  which 
tends  to  draw  it  in  the  direction  of  the  line  AB  ;  since,  by  the 
law  of  inertia  it  would  have  described  the  line  AC=mA  in 
the  time  6,  if  the  second  impulsion  had  not  been  communi- 
cated to  it  ;  but  it  will  describe  the  diagonal  AD  of  the  paral- 
lelogram ABDC.  If  it  should  receive  at  D  a  third  impulse 
capable  of  moving  it  over  the  line  DG  in  a  third  time  6,  it 
will,  for  a  similar  reason,  describe  the  diagonal  DP  of  a 
parallelogram  constructed  upon  DG,  and  DE  the  prolonga- 
tion of  AD,  (fee.  ;  thus,  at  the  end  of  a  time  equal  to  116,  the 
material  point  will  have  described  a  polygon  having  n  sides. 

The  velocity  being  constant  so  long  as  the  material  point 
remains  on  the  same  side  of  the  polygon,  it  follows,  that  if 
at  its  arrival  at  the  extremity  of  either  side,  it  be  not  sub- 
jected to  a  new  impulse,  it  will  continue  to  move  in  the 
direction  of  this  side,  with  a  constant  velocity. 

42S.  If  the  time  6  be  supposed  indefinitely  small,  the  im- 
pulsions will  be  communicated  in  consecutive  instants,  and 
the  polygon  will  then  be  transformed  into  a  curve. 

The  time  é  being  supposed  indefinitely  small,  it  may  be 


CURVILINEAR   MOTION.  211 

represented  by  dt,  and  the  side  of  the  polygon  which  is 
passed  over  in  this  time,  will  become  the  element  of  the 
curve  :  consequently,  to  determine  the  velocity,  which  will 
be  measured  by  the  space  which  the  body  would  pass  over 
in  the  direction  of  the  tangent,  in  a  unit  of  time,  if  the  in- 
cessant force  should  cease  to  communicate  new  impulses, 
we  must  multiply  ds,  the  element  of  the  curve,  by  the  num- 
ber of  times  that  dt  is  contained  in  unity  ;  that  is,  we  mul- 
tiply ds  by—-,  and  we  thus  obtain 
^  ^  ^  dt 

_ds 

~di' 
429.  Let  the  body  be  supposed  to  describe  the  polygon 
m,  711',  m",  m'",  &c.  {Fig.  167),  receiving  increments  to  its 
velocity  at  the  points  7n,  m',  J7i",  m'",  &c.  Let  v,  v',  v",  v'", 
&c.  represent  the  velocities  which  the  body  has  acquired  at 
the  points  m,  m',  m",  m'",  «fcc,  and  ê,  ô',  ê",  ô'",  (fcc.  the  times 
employed  in  describing  the  sides  mm',  m'm",  m"m"',  &c. 
Since  each  of  these  sides  is  supposed  to  be  described  with  a 
constant  velocity,  we  shall  have,  by  the  principles  of  uniform 
motion, 

m.m'=v0,     m'm"=v'6',     m"m"'=v"6",  &c.  ; 
and  the  perimeter  of  the  polygon  will  therefore  be  expressed  by 

vê+v'6'  +  v"6"-^6cc. 
If  we  project  the  sides  of  this  polygon  on  the  co-ordinate  axes, 
denoting  by  «,  /S,  y,  «',  /3',  y',  (fcc.  the  angles  formed  by  the 
sides  mm',  m'm",  m"m"',  &c.  with  these  axes,  the  projections 
of  the  sides  will  be  expressed  by 

v6  cos  «,  v'ô'  cos  «',  v"ô"  cos  «",  <fcc.,  on  the  axis  of  x, 
v6  COS  /3,  v'ê'  cos  jS',  v"6"  cos  jS",  «fcc,  on  the  axis  of  y, 
vê  cos  y,  v'ô'  cos  y',  v"e"  COS  y",  &c.,  on  the  axis  of  z  ; 
and  the  projection  nn'n"n"',  <fcc.  of  the  perimeter  mm'm"m"'^ 
on  the  axis  of  x,  will  be  expressed  by 

v6  cos  ci  +  v'ê'  cos  «,'-f  f'T'  cos  «"-f  &c (178). 

It  thus  appears  that  while  the  material  point  m  describes  the 
polygon  mm'm"m"',  &c.,  its  projection  n  will  describe  the  space 
nn'n"n"',  &c.     But  if  the  point  n  were  merely  solicited  by  a 

02 


212 


DYNAMICS. 


force  X  directed  along  the  axis  of  .r,  and  of  such  intensity 
that  the  point  should  describe  the  spaces  nn',  n'n",  n"n"\  ôùc, 
in  the  times  e,  ê',  6",  &.c.,  with  the  velocities  v  cos  x,  v'  cos  «', 
v"  cos  cc",  &c.,  the  space  passed  over  on  the  axis  of  x  would 
be  expressed  by 

V  cos  u9-\-v'  COS  u'ê'+v"  COS  ct"ê"  +  &,c (179). 

In  obtaining  the  expression  (179),  no  reference  has  been  had 
to  the  components  of  the  velocity  parallel  to  the  axes  of  y 
and  z  ;  and  the  identity  of  the  expressions  (178)  and  (179) 
therefore  proves  that  when  the  point  7n  is  transported  in 
space,  its  projection  moves  on  the  axis  of  x;  as  thougli  the 
other  two  components  of  the  velocity  did  not  exist. 

The  same  remarks  being  applicable  to  the  other  two  axes, 
and  the  polygon  becoming  a  curve  when  the  number  of  its 
sides  is  increased  indefinitely,  it  follows  that  when  a  material 
point  solicited  by  an  incessant  force  describes  a  curve  in 
space,  each  projection  of  the  point  moves  independently  of 
the  motions  of  the  other  two. 

Thus,  by  calling  X,  Y,  and  Z  the  components  of  the  in- 
cessant force  ?>,  parallel  to  the  three  axes,  we  can  regard 
these  components  as  forces  which  impress  on  the  projections 
of  the  material  point  motions  which  are  entirely  independent 
of  each  other. 

430.  To  determine  the  analytical  expressions  for  these 
incessant  forces,  we  remark,  that  while  the  material  point 
describes  the  space  ds,  its  projections  describe  the  spaces  dx, 
dy,  and  dz  respectively  :  the  velocities  of  the  projections  will 

therefore  be  represented  by  — -, -^,  and-- :  and  since  the 

dt    dt  dt 

incessant  force  is  equal  to  the  differential  coefficient  of  the 

velocity  considered  as  a  function  of  the  time,  we  shall  have, 

by  regarding  dt  as  constant, 

^^— X 
'dt^~ 


dt^ 
d'z     „ 


(180) 


CURVILINEAR    MOTION.  213 

Such  are  the  equations  which  serve  to  determine  the  circum- 
stances of  the  motion  of  a  material  point  describing  a  curve. 

431.  When  the  functions  X,  Y,  and  Z  are  given  by  the 
lature  of  the  problem,  and  if  the  integrals  of  the  equations 
^^180)  can  be  obtained,  these  integrals  will  give  three  relations 
between  the  four  variables  x,  y,  z,  and  t  :  the  quantity  t  being 
eliminated,  there  will  remain  two  relations  between  x,  y,  and 
z,  which  will  represent  the  equations  of  the  trajectory,  or 
curve  described  by  the  7naterial  point  under  the  influence  of 
the  incessant  forces. 

When  the  forces  are  situated  in  a  single  plane,  which  may 
be  taken  as  that  of  x,  y,  the  trajectory  will  he  contained  in  the 
same  plane,  and  it  will  then  only  be  necessary  to  use  the  two 
equations 

dt^~~  '  dt^~  ' 
When,  by  the  nature  of  the  problem,  the  quantities  X  and  Y 
are  known,  and  if  the  integrals  of  these  equations  can  be 
obtained,  they  will  contain  no  other  variables  than  x,  y,  and  t  ; 
thus,  by  eliminating  t,  we  shall  find  a  relation  between  x 
and  y,  which  may  be  written  under  the  following  form, 

this  relation  will  be  the  equation  of  the  plane  curve  described 
by  tha  material  point. 

432.  The  velocity  of  the  material  point  at  any  instant  is 
expressed  by 

ds 
dt 
hut  the  element  ds  of  the  arc  of  a  curve  situated  in  space, 
being  considered  as  an  indefinitely  small  right  line,  whose 
projections  on  the  co-ordinate  axes  are  represented  by  dx,  dy, 
and  dz.  the  value  of  this  element  will  be 
^{dx^+dy'^+dz''). 
Substituting  this  value  in  the  preceding  equation,  we  have 

v=\ ^{dx^  ^dy^  +dz.^), 
dt 

or,  since  the  difierentials  are  taken  with  reference  to  ;  as  a 
variable, 


214 


DYNAMICS. 


,;=V^(^^  +  ^>^\ (181). 

The  angles  formed  by  the  direction  of  the  motion  with  the 
co-ordinate  axes  will  result  from  the  equations 

dx 

V  cos  «  =  — -, 

dt' 

dy 

V  cos  &  =  —-, 

^     dt' 

dz 

V  cos  y=-^-. 

dt 

433.  The  velocity  may  likewise  be  determined  in  the 
following  manner.  Let  the  equations  (180)  be  multiplied 
respectively  by  2dx,  2c/y,  and  2dz  ;  the  sum  of  these  pro- 
ducts will  give 

2dx.d^x-\-2dy.d''y  +  2dz.d'z     ^,^,    ,  ^  ,    ,  v-j  ^ 
^ =  2(Xc?x  +  Ydy  +  Zrfz)  : 

and  si^ce  the  first  member  is  the  differential  of  dx'-^-dy" 
•{■dz^t^  divided  by  df^,  we  shall  have 

or,  rejjîacing  dx'  +di/'  +dz-  by  its  value  ds-,  and  integrat- 
ing, wi^Éibtain 

'■Jl         iÇ--m^d.v-hYdy  +  Zdz)  +  C; 

^-  ds 

and  by  substituting  î?  for  — ,  we  find 

v'=2f{Xdx-{-Ydy-{-Zdz)+C (182). 

434.  It  thus  appears  that  the  determination  of  the  velocity 
will  depend  on  the  integration  of  the  expression 

J\Xdx  +  Ydy-hZdz) (183). 

When  this  integration  is  possible,  the  integral  will  be  a 
function  of  the  variables  x,  y,  and  z,  and  the  equation  (182) 
may  be  written  under  the  form 

v"-^2F{x,y,z)-\-C (184). 

To  determine  the  value  of  the  constant,  we  must  know  the 
velocity  of  the  moveable  point,  at  a  given  point  of  the  trajee- 


CURVILINEAR    MOTION.  215 

tory.     Thus,  if  V  be  the  velocity  at  that  point  which  cor- 
responds to  the  co-ordinates  ar=a,  y  =  6,  z=c,  we  shall  have 

The  value  of  C  being  deduced  from  this  equation,  and  sub- 
stituted in  equation  (184),  we  shall  obtain 

v--N-=2F{x,  y,  ^)-2F(a,  6,  c). 
435.  The  expression  (183)  is  integrable  when  the  move- 
able point  is  subjected  to  the  action  of  a  force  which  is 
constantly  directed  towards  a  fixed  centre.  To  demonstrate 
this  proposition,  we  will  represent  the  resultant  R  of  the 
several  forces  acting  on  the  material  point  by  CD,  a  portion 
of  the  line  CM  drawn  from  the  point  to  the  fixed  centre 
{Fig.  168)  ;  let  this  centre  be  assumed  as  the  origin  of  co-ordi- 
nates, and  denote  by  x  the  distance  of  the  point  M  from  the 
origin,  and  by  a,  /3,  y  the  angles  formed  by  CM  with  the  axes 
of  co-ordinates  :  the  direction  of  the  resultant  forming  the 
same  angles,  we  shall  have 

X  =  RC0S«,      Y=RC0S|3,      Z=Rcosy, 
and  consequently 

X      COS»        Y_C0S/3        Z_C0S7  (\QK\ 

COS  /3        Z       COS  y        X      COS  a. 

But  if  X.  y,  and  z  denote  the  co-ordinates  of  the  point  M,  we 
shall  have 

a;=ACOS«,     y=Acos/3,     2;=Acosy; 

whence,  by  division, 

a:_cos«      y cos  (S       r     cos  "/ 

y     cos  /s'     z    cos  y      X     cos  «  ' 

these  values  substituted  in  equations  (185),  give 

yX— .rY=0,    5;Y-yZ=0,     .rZ— ;^X=0. 

If  in  these  equations  we  replace  X,  Y,  and  Z  by  their  vn1nf>«! 

deduced  from  equations  (180),  we  shall  find 

d^x       d^u    _ 

y- :r— ^=0, 

^dr-         dt^        ' 

d^y       d-z     „ 

d' z        d'X     ^ 
x^  - — z-, — =0. 
dt^       dr~ 


216  DYNAMICS. 

Multiplying  the  first  of  these  equations  by  dt,  integrating  and 
reducing,  we  obtain 

ydx-xdy^^ 

dt  ^       ' 

The  other  two  equations  being  treated  in  a  similar  manner, 
we  find 

ydx — xdy  =  Cdt, 

zdy  — ydz = Cdt, 

xdz — zdx= Cdt. 

If  we  multiply  each  of  these  equations  by  the  variable  which 

it  does  not  contain,  and  take  the  sum  of  the  products,  there 

will  result 

d^Cz-\-C'x+C"y)=0, 
or, 

C^  +  C'.r  +  C"y=0. 

This  equation  being  that  of  a  plane  passing  through  the  ori- 
gin of  co-ordinates,  or  centre  of  attraction,  it  follows  that  the 
point  will  describe  a  plane  curve. 

In  the  resolution  of  this  problem  it  will  therefore  be  unne- 

d'^  z 
cessary  to  employ  the  equation  Z=-—,  and  it  will  simply  be 

necessary  to  integrate  the  equation  (186),  which  may  be 
written  thus  : 

ydx — xdy=Cdt\ 

and  from  this  we  deduce 

f{ydx-xdy)=--Ct+C' (187). 

To  determine  the  value  of  this  integral,  we  remark  that  ydx 
being  the  element  of  a  surface  bounded  by  a  curve,  we  can 
suppose  this  surface  to  be  included  within  the  limits  x=0  and 
x=-CV  {Pig.  169);  thus,  the  expression  Tydr  will  be  repre- 
sented by  the  area  LCPM.  If  from  this  area  we  subtract 
the  triangle  CPM,  there  will  remain 

sector  LCM=area   LCPM-triangle  CPM, 
or, 

sector  LCM=yyo?ar—'^; 
differentiating  and  reducing,  we  find 


CURVILINEAR    MOTION.  217 

</(sectorLCM)=^i^:=^; 

and  again  integrating, 

2 .  sector  'LCM.=J{ydx—xdy)  : 
hence>  the  equation  (187)  can  be  reduced  to  the  following: 

2  .  sector  LCM=Ci (188)  ; 

the  constant  C  is  here  suppressed,  since  we  may  always  re» 
gard  the  times  as  reckoned  from  the  instant  when  the  move- 
able point  is  situated  at  the  point  L,  in  which  case  the  sector 
will  become  equal  to  zero. 

If  we  make  C=2A,  the  equation  (188)  will  become 
sector  LCM=A^  ; 
from  which  we  conclude,  that  when  a  material  point  solicited 
by  a  force  which  is  constantly  directed  towards  a  fixed  centre^ 
describes  a  curve  LM  about  this  centre^  the  area  of  the  sector 
LCM  described  by  the  radins  vector  drawn  to  the  material 
point  is  co7istantly  jiroportional  to  the  time  which  the  point 
employs  in  describing  the  curve.  This  property  is  called 
the  principle  of  areas  proportional  to  the  times. 

436.  The  formula  (183)  i.  always  integrable  when  the 
forces  are  directed  towards  fixed  centres,  their  intensities 
being  at  the  same  time  functions  of  the  distances  of  the 
material  point  from  these  centres. 

Let  M  represent  the  place  of  the  material  point  {Fig.  170), 
which  is  attracted  by  the  forces  P,  P',  P",  (fcc.  towards  the 
fixed  centres  C,  C,  C",  &c.  :  denote  by 

A-,  y,  z^  the  co-ordinates  of  the  point  M, 
a,  6,  c,  the  co-ordinates  of  the  centre  C, 
a',  6',  c',  the  co-ordinates  of  the  centre  C, 
a",  6",  c",  the  co-ordinates  of  the  centre  C", 
&c.  (fee.  (fee. 

p,  p\  p'\  (fee,  the  distances  CM,  CM,  C"M,  (fee.  ; 
«,  0,  y,  the  angles  formed  by  p  with  the  axes  of  co-ordinates, 
a',  /3'  y',  the  angles  formed  by  p'  with  the  same  axes, 
«",  |8",  y",  the  angles  formed  by  p"  with  the  same  axes, 
&c.  (fee.  (fee.  (fee. 

The  total  resultant  of  the  attractive  forces  will  have  the  fol- 
lowing components  parallel  to  the  three  axes, 

19 


218  DYNAMICS. 

X=Pcos«4-P'cos«'+P"cos«"  +  (fcc.  ^ 

Y=P  cos  /3+P'  COS  /s'  +  P"  COS  0"  +  (fee.  V (189). 

Z  =  P  cos  y  +  F  COS  y'  +  P"  COS  y "  +  (fcc.  ^ 

The  projection  of  the  right  Hue  CM  on  the  axis  of  .r  being  rep- 
resented by  BD  {Fig.  170),  we  have 

BD=AB-AD; 
and  by  observing  that  AB  and  AD  are  the  co-ordinales  x  and 
a  of  the  points  M  and  C,  and  that  BD,  being  the  projection 
of  MC  on  the  axis  of  x,  is  expressed  by  j)  cos  «,  we  shall  find, 
by  substituting  these  values  in  the  preceding  equation, 

p  cos  ct=x — a  ; 
the  same  remarks  being  applicable  to  the  projections  on  the 
other  two  axes,  we  shall  have 

2^  cos  u^=x — a,    J)  cos/3=y — h,     j)  cosy=2r — c 
And  in  like  manner, 
p'  cos  oc'=x — a',    p'  cos  i3'=2/ — b',    p'  cos  y'=.z — c', 
J)"  cos  ct"=x  —  a",    jj"  cos  (i"=y  —  b'\    p"  cos  y"=-z — c", 
&c.  &c.  (fee. 

By  eliminating  the  cosines  of  these  angles,  the  equations 
(189)  become 

X=P^^^+P'^^+P"^^'+&c., 
p  p'  p" 

Y=pfc-VP'^+F'^  +  &c., 
J)  J)  p 

Z^-p^^+F'^^+V^^  +  acc. 
p  p'  p' 

These  values  substituted  in  the  formula  (183)  give 
f{Xdx+Ydy  +  Zdz)=^fp{^^^dx-\-'^^dy+''-^dz\ 

+(fec.  (fee.  (fee (190). 

But  the  distances  of  the  point  M  from  the  centres  C,  C,  C", 
(fee.  being  given  by  the  equations 

{x-aY->r{y—hy-{-{z-cY=p^, 

&c.  (fee.  (fee, 


CURVILINEAR    MOTION.  219 

we  shall  obtain,  by  differentiating, 

dx-\-- ay-\ az=ap, 

P  P  P 

ax  •\-- dy-\ dz = dp . 

p'  P  P' 

&c.  &c.  &c.  ; 

and  substituting  these  values  in  equation  (190),  we  find 
J{Xdx-\-Ydy->rZdz)^f{Vdp-^V'dp'  +  V"dp"^&cc.)....{l<èl). 
But  the  forces  P,  P',  P",  (fee.  are,  by  hypothesis,  functions  of 
the  distances  jo,  p',  p",  (fee;  the  expression  Vdp-\-Vdp'-\- 
V'dp"  will  therefore  contain  but  a  single  variable  in  each 
term,  and  its  integral  may  be  effected  by  the  method  of 
quadratures. 

It  should  be  observed  that  the  factors  dp,  dp',  dp",  (fee. 
may  become  negative,  if  the  expressions  x — a,  y — h,  z — c, 
X — o',  (fee.  should  be  transformed  into  a—x,  b—y,  c—z, 
a'—x,  (fee. 

437.  For  the  purpose  of  making  an  application  of  the  pre- 
ceding theorem,  let  it  be  required  to  determine  the  velocity 
of  a  material  point  which  moves  from  rest,  under  the  influ- 
ence of  a  force  of  attraction  which  is  constantly  directed 
towards  a  fixed  centre,  and  which  varies  in  intensity  in  the 
inverse  ratio  of  the  square  of  the  distance  from  the  position 
of  the  point  to  the  fixed  centre.  Let  the  direction  of  the 
force  be  supposed  to  coincide  with  the  axis  of  z  :  the  co- 
ordinate axes  being  disposed  as  in  Pig.  171,  the  intensity  of 
the  force  and  the  co-ordinate  z  will  increase  together,  and  we 
shall  have 

p=AG—AM=c—z,     dp——dz. 

If  g  represent  the  intensity  of  the  force  at  the  distance  r  from 
the  centre  C,  and  P  its  intensity  at  the  distance  p,  we  shall 
have  the  proportion 

P    .  1      1. 

^■^''■7^''p^^ 
whence, 

but  dp  being  negative,  the  quantity  Vdp  should  be  replaced 


28d  DYNAMICS. 

by  —- — df  ;  integrating,  we  reduce  the  equation  (191)  to 

r 

This  vahie  being  substituted  in  formula  (182)  gives 

v2-?ll!4-C (192). 

V 
To  determine  the  value  of  the  constant  C,  we  suppose  the 
body  to  commence  its  motion  at  a  point  whose  distance  from 
the  centre  of  attraction  is  represented  by  a  ;  the  velocity  at 
this  point  being  equal  to  zero,  we  have 

a 
or, 

2g^r2 


0= 


the  equation  (192)  will  therefore  become 
..=2^r»(i_i). 

If  a  be  regarded  as  the  unit  of  distance,  the  value  of  v^  will 
become  identical  with  that  determined  in  Art.  409. 

438.  To  apply  the  formulas  (180)  we  will  first  investigate 
the  trajectory  described  by  a  material  point  which  moves 
under  the  influence  of  a  single  impulse.  In  this  case,  the 
incessant  forces  being  equal  to  zero,  we  shall  have 

X=0,     Y=0,     Z=0; 
and  the  equations  (180)  reduce  to 

^=0     ^-0     '-^=0^ 

multiplying  by  dt^  they  become 

^=0     ^=0     — =0 
dt        '      dt        '      dt 

The  integrals  of  these  equations  are 

dx  dy  dz 

-T-=a,     -~=h,     -i-=c (193). 

dt        '     dt        '     dt  ^       ^ 

Substituting  these  values  in  equation  (181),  we  find 


MOTION    UPON    A   GIVEN   CURVE.  221 

v=^{a^  +b' -\-c')=Si  constant; 
and  denoting  this  constant  by  A,  we  have 

ds     . 
dt        ' 
consequently, 

s=At-{-B] 
and  the  motion  of  the  material  point  will  be  uniform. 

The  motion  is  likewise  rectilinear  ;  for  the  equations  (193) 
give,  by  integration, 

x=at-{-a\     y=ht-\-b\     z=ct-\-c', 
whence,  by  eliminating  t, 

az  ,  a'c — ac'  bz  ,  b'c — be' 

c  c      ^    ^      c  c 

These  equations  evidently  appertain  to  the  projections  of  a 
right  line  on  the  planes  of  x,  z  and  y,  z. 

Of  the  Motion  of  a  Material  Point  when  compelled  to 
describe  a  partictdar  Curve. 

439.  When  a  material  point  m,  without  weight,  has  received 
an  impulse  K  (Fig.  172),  and  is  subjected  to  the  condition  of 
moving  upon  a  particular  curve,  we  can  resolve  this  impulse 
into  two  components,  one  mN=K'  normal  to  the  curve,  the 
other  mT=K"  in  the  direction  of  the  tangent  :  the  normal 
force  will  be  destroyed  by  the  resistance  of  the  curve,  and  the 
tangential  component  will  produce  its  entire  effect  in  com- 
municating motion  to  the  material  point. 

If  we  regard  the  curve  as  a  polygon  mm'm"m'",  &c.  {Fig. 
173),  having  an  infinite  number  of  sides,  the  angle  tm'm" 
formed  by  the  prolongation  of  the  side  mm'  with  the  consecu- 
tive side  m'm"  is  called  the  angle  of  contact  ;  it  will  be 
denoted  by  «  ;  the  plane  tm'm"  is  the  osculatory  plane  at  the 
point  m',  and  in  plane  curves  coincides  with  the  plane  of  the 
curve. 

The  material  point  m,  being  solicited  by  a  force  K,  receives 
a  primitive  velocity  v,  causing  it  to  describe  the  side  mm'  ; 
but  having  arrived  at  the  point  m',  it  is  deflected  from  its 
course,  and  describes  the  side  ?n'm".    By  this  deflection  it 


^^ 


DYNAMICS. 


necessarily  undergoes  a  loss  of  velocity  which  will  now  be 
estimated. 

For  this  purpose,  let  the  velocity  v  be  represented  by  the 
line  m'q.  This  velocity  being  resolved  into  two  components 
9?i'n  and  m'l,  respectively  parallel  and  perpendicular  to  the 
side  7ii'j}i",  we  shall  have 

m' 1=971' q .  sin  tm'm",    'm'n=m,'q  .  cos  Vnilml'^ 
or, 

m'l=v .  sin  »,  tn'n^v .  cos  t». 
The  component  v .  sin  »  being  destroyed  by  the  resistance  of 
the  polygon,  the  velocity  v  will  be  reduced  to  v .  cos  a  ;  and 
consequently,  the  velocity  lost,  being  equal  to  the  primitive 
velocity  diminished  by  the  velocity  actually  remaining,  will 
be  expressed  by  ^(l— cos  i"). 

When  the  polygon  is  supposed  to  become  a  curve,  the 
angle  tm'm"  becomes  infinitely  small,  and  the  quantity 
v(l — cos  u)  is  at  the  same  time  an  infinitely  small  quantity 
of  the  second  order. 

To  prove  that  this  is  the  case,  we  observe  that  1 — cos» 
represents  the  versed  sine  DB  of  an  angle  »  {F^g-  17^4), 
measured  by  the  arc  BC  ;  and  we  have  the  proportion 

AD  :  CD  :  :  CD  :  DB. 
But  when  the  arc  CB  becomes  infinitely  small,  CD  will  be  so 
likewise  ;  and  since  CD  is  then  infinitely  small  with  respect 
to  AD,  it  follows  from  the  above  proportion,  that  DB  must 
be  infinitely  small  with  respect  to  CD,  or  that  it  is  an  infi- 
nitely small  quantity  of  the  second  order.  Thus,  the  velocity 
lost  at  each  side  ot  the  polygon  being  an  infinitely  small 
quantity  of  the  second  order,  it  may  be  neglected,  since  the 
sum  of  these  velocities,  although  infinite  in  number,  will  con- 
stitute but  an  infinitely  small  quantity  of  the  first  order,  which 
may  be  neglected  in  comparison  with  the  original  velocity  v. 
Hence,  we  conclude,  that  a  material  point  which  is  compelled 
to  describe  a  curve,  preserves  undiminished  the  velocity 
which  was  originally  communicated  to  it. 

440.  The  component  of  the  velocity  v .  sin  a  with  which 
the  material  point  is  pressed  against  the  curve,  and  which  is 
destroyed  by  the  curve's  resistance,  varies  constantly  as  the 


MOTION    UPON    A    GIVEN    CURVE.  223 

point  changes  its  position,  since  sin  *  is  constantly  variable  : 
we  may  regard  this  resistance  exerted  by  the  curve  as  an 
incessant  force  constantly  acting  upon  the  point  and  deflecting 
it  from  the  tangent  along  which  it  would  otherwise  tend  to 
move. 

When  there  are  several  forces  acting  on  the  material  point, 
we  resolve  each  in  a  similar  manner,  and  the  sum  of  the  nor- 
mal components  must  then  be  added  to  the  pressure  arising 
from  the  component  of  the  velocity. 

441.  Let  it  be  supposed  that  a  force  N  equal  and  directly 
opposed  to  the  resultant  of  all  the  normal  forces  is  applied  to 
the  material  point  :  this  force  will  produce  precisely  the  same 
effect  as  the  resistance  offered  by  the  curve,  and  the  latter  will 
therefore  be  represented  by  N.  Let  «,  /s,  y  be  the  angles 
formed  by  the  direction  of  the  force  N  with  the  co-ordinate 
axes  ;  the  components  of  N  in  the  direction  of  the  axes  will- 
be  respectively 

N  cos  «,     N  cos  /3,     N  cos  y, 
and  should  be  added  to  the  components  of  the  incessant  forces 
in  the  general  equations  of  motion  (180)  :  we  shall  thus  obtain 

- — =X4-N  cos  « 

^=Y+N  cos  ,3  ^ (194). 

- — =Z+N  cos  y 

dp 

To  these  equations  may  be  added  two  others  which  result 
from  the  relations  existing  between  the  angles  u,  /3,  and  y  ; 
the  first  of  these  equations  is 

cos-  «+C0S2  /3-{-COS2  y=l (195). 

The  second  is 

cos  «  .  cos  a'-f-COS  /3  .  COS  /3'-fC0S  y  .  COSy'  =  0, 

in  which  »,  /3',  y'  represent  the  angles  formed  by  the  tano-ent 
to  the  curve  with  the  co-ordinate  axes.  The  cosines  of  these 
last  angles  may  be  expressed  as  follows  : 

,    d.v  ,    dy  ,     dz 

cos  «'=--,      C0S/3'=-/,      COSy'=--: 

as  as  ds 


224  DYNAMICS. 

these  values  substituted  in  the  preceding  equation  convert  it 
into 

-^  cos  u-{-^  cos  /3+-^  cos  y=0 (196). 

as  as  its 

442.  To  determine  the  velocity  of  the  material  point,  let 
the  equations  (194)  be  multiplied  respectively  by  2dx,  2dy, 
and  2dz  :  their  sum  being  taken,  we  shall  obtain 

2dx'^+2dy^  +  2dz^^2(Ldx^Ydy+Zdz) 
dt''  dt^  dP 

4-2N(c?a; .  cos  a+f/y  .  cos  ^-l-dz  .  cos  y)  : 

the  last  term  of  this  equation  being  equal  to  zero,  by  formula 

(196),  there  remains 

2dx'^-V2dy'^+2dz'^^=2{X.dx-^Ydy+Zdz)  ; 
or, 

d{dx^+dr  'rdz^)^2{Xdx^Ydy-^Zdz)  : 

whence,  by  substitution  and  integration,  we  find 
1^  =2f{Xdx+Ydy^Zdz)+C  ; 

or, 

v'=2f{Xdx-^Ydy+Zdz)+C (197). 

443.  When  the  material  point  merely  receives  an  impulse, 
without  being  acted  upon  by  incessant  forces,  we  have 

X=0,     Y=0,     Z=0; 
and  consequently, 

v'  =a  constant. 
Thus,  when  the  material  point  is  compelled  to  describe  a 
curve,  being  acted  upon  only  by  an  impulse,  its  velocity  will 
remain  invariable.  This  result  accords  with  that  which  has 
been  already  obtained  (Art.  438),  on  the  supposition  that  the 
motion  is  perfectly  free. 

444.  Let  the  material  point  which  is  supposed  to  describe 
the  curve,  be  acted  on  by  the  forceof  gravity  ;  we  shall  then 
have 

X-0,     Y=0,     Z=^; 
and  the  equation  (1 97)  will  be  reduced  to 
v''=2fgdz-{-C. 


MOTION   UPON    A    GIVEN    CURVE.  229 

If  the  velocity  v  be  supposed  equal  to  V,  when  «=0,  we  shall 
find 

This  value  substituted  in  the  preceding  equation  gives 

whence, 

v=^{2gz+N^) (198). 

This  expression  for  the  velocity  being  independent  of  the 

relations  which  may  exist  between  the  co-ordinates  x,  y,  and  z^ 

it  follows  that  the  velocity  will  be  the  same  for  the  same 

value  of  z,  whatever  may  be  the  form  of  the  curve. 

To  determine  the  expression  for  the  time  employed  by  the 

material  point  in  describing  a  given  portion  of  the  curve,  we 

ds 
replace  v  by  its  value  -r^,  and  thus  obtain 


whence, 


<"=7(^JW,---<^«^>^ 


or,  by  substitution, 

^'^       VC^gz  +  Y^)        (''^^- 

To  integrate  this  equation,  it  will  be  necessary  to  reduce  it,  by 
means  of  the  equations  of  the  curve,  to  one  which  shall  con- 
tain but  two  variables  ;  thus,  if  the  equations  of  the  curve 
are 

f{x,z)=0,    At/,z)=^ (201), 

we  may,  by  the  aid  of  these  equations,  in  connexion  with 
equation  (200),  eliminate  two  of  the  three  variables  x,  y,  and 
z  ;  and  it  will  then  only  be  necessary  to  integrate  an  equation 
expressing  the  relation  between  dt  and  one  of  the  co-ordi- 
nates of  the  moveable  point. 

445.  If,  for  example,  the  curve  be  supposed  to  become  a 
right  line,  the  equations  (201)  will  be  of  the  form 

x=az+u,     y=bz+^ (202): 

from  which  we  deduce 

dx^adz,    dy=bdz\ 
P 


226  DYNAMICS. 

and  by  substituting  these  values  in  the  formula  (200),  it  is 
transformed  into 

dzy{l  +  a'-\-b') 

If  the  point  be  supposed  to  move  from  rest,  its  initial  velocity 
V  will  be  equal  to  zero,  and  we  shall  have,  by  division, 
dt  dz 

whence,  by  integration, 

^  ]■^/{2gz) (203). 


The  constant  introduced  by  integration  becomes  equal  to  zero, 
since,  by  hypothesis,  when  ^=0,  -y^O,  and  z--0  (Art.  444). 

446.  To  determine  the  space  passed  over  in  the  time  t,  we 
suppress  V  in  equation  (199),  which  then  becomes 

and  eliminating  z  by  means  of  equation  (203),  there  results 
,  g-t  .dt 

and  by  integration, 

10-/3 

9—  "-  4-C  • 

which  proves  that  the  motion  is  similar  to  that  of  a  body  on 
an  inclined  plane,  as  might  have  been  anticipated. 

447.  The  co-ordinates  .r,  y,  and  z  are  readily  determined 
in  functions  of  the  time  ;  for  the  latter  is  given  by  formula 
(203),  and  this,  taken  in  connexion  with  equations  (202),  will 
determine  a;  and  y  in  functions  of  t. 

448.  If,  as  in  the  present  instance,  the  point  be  supposed 
to  describe  a  plane  curve,  and  if  the  incessant  forces  act  en- 
tirely in  this  plane,  we  may,  by  placing  the  axes  of  t  and  y 
in  this  plane,  dispense  with  the  consideration  of  the  third  of 
equations  (194)  ;  the  formulas  (195)  and  (196)  will  then  be 
reduced  to 

cos'«-l-cos2j3=l,     ^-cos«4— /cos/3=0; 
as  as 


MOTION    UPON    A    GIVEN    CURVE.  227 

and  the  two  equations  of  the  curve  will  be  replaced  by  the 
single  relation 

449.  The  velocity  being  given  by  formula  (198),  without 
the  aid  of  equations  (201),  we  conclude  that  the  velocity  ac- 
quired by  the  moveable  point  is  independent  of  the  form  of 
the  curve,  being  determined  by  the  value  of  the  vertical  ordi- 
nate. Consequently,  if  from  the  point  O  {Fig.  175),  where 
^=0,  and  v— V,  we  draw  the  arcs  of  different  curves  OM, 
OM',  OM",  (fcc,  terminated  by  the  horizontal  plane  KL,  the 
ordinates  z  of  the  first  and  last  points  of  all  these  arcs  being 
equal,  it  follows  that  different  bodies  departing  from  the  point 
O  with  the  common  velocity  V,  and  describing  the  several 
curves,  will  all  have  acquired  the  same  velocity  when  they 
shall  have  arrived  at  the  points  M,  M,'  M",  (fee,  situated  in  the 
same  horizontal  plane. 

450.  In  general,  whatever  may  be  the  number  of  forces 
acting  on  the  moveable  point,  if  the  equation  (197)  be  inte- 
grable,  we  can  determine  the  velocity  v  without  knowing 
the  nature  of  the  curve  described.  For,  the  values  of  the  in- 
cessant forces  X,  Y,  and  Z,  expressed  in  functions  of  the  co- 
ordinates X,  y,  and  z.  being  substituted  in  equation  (197),  if 
the  expression 

f{X.dx  +  Ydy  +  Zdz) 
becomes  integrable,  we  may  represent  it  by 

/(^j  y,  z)  ; 
and  the  equation  (197)  will  then  reduce  to 

'v'=2f{x,y,z)+C. 
If  we  denote  by  a,  h,  and  c  the  values  of  x,  y,  and  z  which 
correspond  to  the  velocity  V,  the  value  of  C  will  become 
known  ;  thus, 

C=V2-2/(«,  6,c); 

and  replacing  C  by  this  value  in  the  general  expression  for 
the  velocity,  we  find 

v^  =V*  -\-2f{x,  y,  z)-2f{a,  b,  c)  ; 
an  expression  which  depends  only  on  the  initial  velocity,  and 
the  co-ordinates  of  the  first  and  last   points  of  the  curve 

described. 

P2 


228  DYNAMICS. 

451.  It  has  been  explained  (Art.  440)  that  the  normal 
pressure  exerted  against  the  curve  arises  in  part  from  the  nor- 
mal components  of  the  incessant  forces,  and  partly  from  the 
normal  force  due  to  the  velocity.  To  determine  the  value  of 
the  latter,  let  perpendiculars  on  and  o?i'  be  erected  at  the 
middle  points  of  the  equal  consecutive  sides  wtwt'  and  m'tn" 
(FS,g.  176)  of  the  polygon  having  an  infinite  number  of 
sides  :  the  angle  tm'm,",  formed  by  one  of  these  sides  with  the 
prolongation  of  the  other,  W\\\  be  the  angle  which  we  have 
represented  by  •».  But  the  angles  n  and  n'  being  right  anglesj 
we  have 

non'  ■\-  ii'm'n' = 180° = tm'm"  -\-  nm!iï  ; 

and  therefore, 

trn'm" = » = non' = 2nom'. 
The  angle  nom'  being  infinitely  small,  its  sine  may  be  re- 
garded as  equal  to  the  arc  which  measures  it  ;  but  this  sine 

,  ,     m'ît'       m'n'     .  j      ,  i 

IS  expressed  by ,  or ,  smce  7io  and  mo  may  be  con- 

m'o  710 

sidered  equal  ;  hence, 

2m'n     mm' 

û/= = . 

710  710 

If  we  now  return  to  the  consideration  of  the  curve  which  is 
the  limit  of  the  polygon,  the  side  mw'  becomes  the  element 
of  the  curve,  and  7io  the  radius  of  curvature  :  the  preceding 
relation  will  therefore  be  transformed  into 

ds 

0,=  —  , 

y 
y  denoting  the  radius  of  curvature. 

Let  ç>  denote  the  intensity  of  the  incessant  force  which  is 
due  to  the  normal  component  of  the  velocity  :  this  intensity 
being  in  general  expressed  by  the  quotient  of  the  element  of 
the  velocity,  divided  by  the  element  of  the  time,  and  the  ele- 
ment of  the  velocity  being  represented  in  the  present  instance 
by  V .  sin  «,  we  shall  have 

V .sin  a 

or,  since  the  infinitely  small  arc  may  be  substituted  for  its 
sine,  this  expression  becomes 


MOTION  UPON  A  CURVED  SURFACE.         229 
Vu 

replacing  «  by  its  value  found  above,  we  have 

vds  v^ 

^  =  -^'    or^=— . 
yat  y 

The  normal  pressure  resulting  from  the  other  forces  may  be 
determined  by  the  parallelogram  of  forces,  and  this  pressure 
must  then  be  combined  with  that  due  to  the  velocity,  in  order 
to  obtain  the  total  pressure, 

452.  Let  it  be  supposed,  for  example,  that  the  material 
point  describes  a  plane  curve,  and  that  the  incessant  forces 
are  directed  in  the  plane  of  this  curve  :  let  these  forces  be 
reduced  to  their  resultant  R,  and  denote  by  6  the  angle  formed 
by  the  direction  of  the  resultant  with  that  part  of  the  normal 
which  lies  on  the  concave  side  of  the  curve  :  the  component 
of  the  resultant  in  the  direction  of  the  normal  will  be  ex- 
pressed by  R  cos  ^,  and  will  act  in  the  same  or  in  a  contrary 
direction  to  the  pressure  arising  from  the  velocity,  according 
as  6  is  obtuse  or  acute.  The  pressure  arising  from  the  velo- 
city being  always  directed /rowi  the  centre  of  curvature,  the 
entire  pressure  will  be  expressed  by 

N=— — RcosO; 
y 
this  pressure  will  be  exerted /rom  the  centre  of  curvature  so 
long  as  the  quantity  N  is  positive,  and  towards  the  centre  in 
the  contrary  case. 

Of  the  Motion  of  a  material  Point  when  compelled  to  move 
itpon  a  Curved  Surface. 

453.  When  a  material  point  which  is  compelled  to  move 
upon  a  curved  surface  is  subjected  to  the  action  of  inces- 
sant forces,  these  forces,  and  that  resulting  from  the  velocity 
of  the  point,  will  exert  a  pressure  against  the  surface,  which 
will  be  counteracted  by  the  resistance  of  the  surface.  If  we 
denote  this  resistance  by  N,  the  material  point  may  be  re- 
garded as  moving  freely  in  space,  provided  we  include  the 
components  of  the  force  N  in  the  general  equations  (180), 
which  express  the  circumstances  of  motion  of  a  point  under 

20 


230 


DYNAMICS. 


the  influence  of  incessant  forces.  Let  «,  /3,  and  y  represent 
the  angles  formed  by  the  direction  of  the  force  N  with  the 
co-ordinate  axes  ;  its  components  in  the  directions  of  these 
axes  will  be  expressed  by  N  cos  «,  N  cos  /3,  and  N  cos  y  :  con- 
sequently, the  equations  of  the  required  motion  will  be 


- — =X4-N  cos  « 

^=:Y  +  NC0S/3 

^=Z+1S  cosy 
dt^ 


(204). 


The  angles  «,  /2,  and  y  will  become  known  when  the  equation 
of  the  surface  L=0  is  given,  for  we  have,  (Art.  62), 

dx 
cos  ct—  ± 


^m^m^it) 


cos  /3=± 


dy 
dL 
dy 


cos  y=  ± 


dL 

d^ 


^{Èï-m'HÈ) 


the  double  signs  prefixed  to  the  values  of  the  cosines,  indi- 
cate that  they  may  refer  to  the  direction  of  a  force  which 
tends  to  pull  the  point,  either  along  the  normal  to  the  surface, 
or  along  its  prolongation. 
If  we  put,  for  brevity, 

1 

± —  -    — =Y, 


^{Èï^Èï^Èy 


dy 

the  preceding  equations  will  become 
.dL 


cos  «=V 


dx' 


COS/S: 


dy^ 


COS»/— Y 


M. 


dz 


MOTION  UPON  A  CURVED  SURFACE,         231 

these  values  substituted  in  equations  (204),  reduce  them  to 


dt''  dz 


(205). 


If  N  be  ehminated  between  these  equations,  V  will  likewise 
disappear,  and  we  shall  thus  obtain  two  relations,  which, 
taken  in  connexion  with  the  equation  of  the  surface  L=0, 
will  determine  the  co-ordinates  of  the  moveable  point  in 
functions  of  the  time. 

454.  As  an  example  : — Let  it  he  required  to  determine  the 
circumstances  of  the  motion  of  a  material  point  on  the 
surface  of  a  sphere  :  let  the  origin  of  co-ordinates  be  as- 
sumed at  the  centre,  the  plane  of  x,  y  being  horizontal,  and 
the  co-ordinates  z  being  reckoned  positive  downwards  ;  these 
co-ordinates  will  then  be  affected  with  the  same  sign  as  the 
force  of  gravity. 

The  equation  of  the  sphere  being 

lj=x^-\-y^-\-z'—a^=Q (206), 

we  obtain  by  differentiation, 

dl.=2xdx-^2ydy^-2zdz=0 (207), 

and  consequently, 

g-=2x,    ^=2y,    -=2z. 

y      ,  1  ,\. 

~v'(4a^ +4^2+42^)-"^  2a' 

or, 

COS«=±-,     008/3=  ±?^,     cosy=±- (207  a). 

a  a  a 

Again,  the  force  of  gravity  being  the  only  incessant  force 
acting  on  the  material  point,  we  have 

X=0,     Y-0,     Z=g; 
these  values  reduce  the  equations  (205)  to 


232  DYNAMICS. 

^=±N?,    !^=±N^,    '^.  =  ±^-+e (208). 

at-  a       at-  a       at^  a 

The  positive  signs  should  be  taken  together,  and  evidently 
correspond  to  Hke  signs  in  the  vahies  of  the  cosines  of  «,  /î) 
and  y  ;  a  similar  remark  is  apphcable  to  the  negative  signs. 

We  ehminate  ±  N  between  the  two  first  of  these  equations, 
by  miihiplying  them  respectively  by  y  and  i\  and  taking 
their  difference  ;  we  thus  obtain,  after  muhiplying  by  dt^ 

yd^x—xd^y_^    or   ^(y^-^~^^y)-o . 
dt  '  dt  ' 

whence,  by  integration,  dt  being  regarded  as  constant, 

ydx-xdy=Cdt (209). 

A  second  relation  between  the  variables  may  be  found  by 
multiplying  each  ot  the  equations  (208)  by  the  differential  of 
the  variable  which  it  contains  ;  the  sum  of  these  products  will 
give 

dx.d-ix-\rdy .d^'i/-\-dz .d^z      ,  N,    ,    ,     ,    ,      ,  .  ,      , 
dP     ^     -{xdx-\-ydy-\-zdz)-\-gdz  ; 

and  since  the  quantity  included  within  the  brackets  is 
equal  to  zero,  by  equation  (207),  the  preceding  result  will  be 
reduced  to 

dx  .d~x-\-dy  .d^y-\-dz  .  d^z _     , 

W^  ~^  ^' 

multiplying  by  2,  and  integrating,  we  have 

^.^^l±^p:^=2gz+G' (210). 

If  two  of  the  three  variables  .t,  y,  and  z  be  eliminated  between 
the  relations  (206),  (209),  and  (210),  ihe  result  will  be  an 
equation  which,  being  integrated,  will  give  a  relation  between 
the  third  co-ordinate  and  the  time  /  :  this  result  will  evidently 
be  independent  of  the  normal  force,  which  has  already  disap- 
peared from  these  three  equations. 

455.  The  equations  (207)  and  (209)  being  squared,  give 

x^dx-+  2xydxdy  +  y""  dy^  =z^  dz^^ 

y^  dx^  —  2xydxdy  +  x^  dy- =C^  dt^ . 

The  sum  of  these  equations  being  taken,  the  middle  terms  of 

the  first  members  will  disappear,  and  we  shall  have 


MOTION  UPON  A  CURVED  SURFACE.        233 

substituting  for  {x^  +y")  its  value  deduced  from  equation 
(206),  there  results 

,  ,  ,   ,  ,     C^dP-ifZ^dz^ 
dx^'  -\-dy= —  ; 

and  eliminating  dx'^  -\-dy^  between  this  equation  and  (210), 
we  find 

dt= — (211). 

V'[(a^-z^)(2^z  +  C')-C^l  ^      ^ 

The  integral  of  this  equation,  which  can  only  be  obtained  by- 
approximation,  will  give  the  value  of  z  in  functions  of  the 
time. 

456.  To  determine  the  expressions  for  the  other  co-ordinates 
in  functions  of  the  time,  we  will  suppose  ft  to  represent  the 
approximate  value  of  z  determined  from  the  integration  of 
the  preceding  equation  :  if  this  value  be  substituted  in  equation 
(210),  we  may,  by  combining  the  resulting  equation  with  that 
designated  as  (209),  obtain  two  relations,  the  first  between  x 
£md  t,  the  second  between  y  and  t  :  but  as  the  variables  in 
each  of  these  equations  would  not  be  separated  by  this  pro- 
cess, we  adopt  another  method  of  determining  the  values  of 
x  and  y  in  functions  of  t. 

Let  KC=x,  DC=y,  iiiD^z  {Fig.  177)  be  the  three  co- 
ordinates of  the  point  tn  on  the  surface  of  the  sphere  ;  if  for  a 
given  value  of  z,  the  angle  CAD,  formed  by  the  projection 
AD  of  the  right  line  AM  with  the  axis  of  x,  were  known,  the 
corresponding  values  of  x  and  y  might  be  readily  determined 
in  functions  of  z.  For,  the  angle  CAD  being  denoted  by  6, 
and  the  radius  Am  by  a,  we  shall  have  KD=^{a^ —z^)] 
and  the  triangle  ACD  right-angled  at  C,  will  give 

AC=AD .  cos  CAD,     CD=AD .  sin  CAD  ; 
or, 

x=^y/{a^—z'')  Xcos  Ô,     y=^{aP-  -z^)  Xsin  0 (212). 

These  two  equations  establishing  a  relation  between  x,  y,  and 
z,  may  be  considered  as  replacing  the  equation  of  the  sphere, 
which  can  be  obtained  by  taking  the  sum  of  their  squares. 
An  additional  variable  6  is  here  introduced,  but  the  number  of 
relations  is  likewise  increased  by  unity. 


234  DYNAMICS. 


From  the  equations  (212)  we  obtain  by  differentiation, 

...(213): 


zdLz 

dx=—s'med6^(a'' — z") — ,cos  Ô 


zdz 
di/=cos  eds^ia' — z^) r- rMn  6 

multiplying  the  first  of  equations  (213)  by  the  second  of 
(212),  and  the  second  of  (213)  by  the  first  of  (212),  and  taking 
their  difl'erence,  we  obtain 

]/dx—xdy=~-{a^—z''){sm''ô  +  cos^6)d6', 
or, 

pdx  —xdy={z^  — a^)d6. 
This  value  substituted  in  equation  (209),  gives 

{z''—a^)d6=Cdti 
and  consequently, 

z^  -a"  ' 
or,  replacing  dt  by  its  value  deduced  from  equation  (211),  we 
obtain 

,  a.C  .dz 

-\z^-a^^)^[{a--z')[2gz+C')-C^] 
This  equation,  being  integrated  by  approximation,  will  deter- 
mine the  value  of  6  :  we  thence  deduce  the  values  of  cos  ê, 
and  sin  6,  which  substituted  in  equations  (212),  determine  the 
values  of  x  and  y  in  functions  of  z,  and  consequently  in 
functions  of  the  time  t. 

457.  The  equation  (210)  proves  that  the  velocity  is  inde- 
pendent of  the  normal  pressure  ;  for,  we  deduce  f  om  that 
equation, 


or, 


*»  ='°^+*^> 


v^=2gz-^0: 


and  consequently, 

v=^{2gz-[-C'). 

To  determine  the  value  of  the  normal  pressure,  we  must 
recur  to  equations  (208)  :  these  being  multiplied  respectively 
by  X,  y,  and  z,  and  added,  give 


MOTION  UPON  A  CURVED  SURFACE.        236 

^,J    =  ±-{x'-^y^+z')+gz (214). 

But  the  diflerential  of  equation  (207),  xdx-\-ydi/  +  zdz=0, 
being  taken,  we  find 

xd''x+yd''i/-i-zd''z_     dx^  ■\-dy''  -\-dz^  __  j, 
dT^  df^ 

and  this  value  substituted  in  equation  (214)  gives,  after  re- 
placing .r»  -^-y^  +2;2  by  «3, 

or, 

a       a 

458.  If  the  moveable  point  be  supposed  situated  at  any 
instant  below  the  horizontal  plane  passing  through  the 
centre  of  the  sphere,  the  ordinate  z  will  be  positive,  and  the 
value  of  ±  N  becomes  negative  :  and  since  N,  which  denotes 
the  intensity  of  a  force,  is  by  hypothesis  an  essentially  posi- 
tive quantity,  the  inferior  sign  must  be  taken  in  order  that 
— N  may  be  essentially  negative.  Hence,  it  will  be  neces- 
sary to  take  the  inferior  signs  in  equations  (208),  and  also  in 
equations  (207  a).  The  resistance  of  the  surface  will  there- 
fore be  directed  towards  the  centre,  or  the  material  point  must 
be  regarded  as  situated  upon  the  concave  surface  of  the  sphere. 

When  the  material  point  rises  above  the  horizontal  plane 
of  a-,  y,  the  ordinate  z  will  become  negative,  and  the  quantity 
— tj3 — gz  may  then  become  positive.  In  such  case,  the 
superior  signs  must  be  taken  in  equations  (207  a)  and  (208), 
and  the  resistance  of  the  surface  will  be  exerted  from  the  cen- 
tre, or  the  body  must  be  supposed  to  be  on  the  convex  surface. 

The  pressure  exerted  against  the  surface  will  be  equal  and 
directly  opposed  to  the  resistance  which  it  offers,  and  will 

therefore  be  represented  by ^—  without  reference  to  the 

sign  of  z. 

If  the  moveable  point  be  retained  upon  the  surface  of  the 
sphere  by  an  inflexible  thread  connecting  the  point  with  the 
centre,  this  thread  will  experience  a  tension  so  long  as  v^  -\-gz 
is  positive  ;  but,  on  the  contrary,  there  will  be  a  tendency  to 
compress  the  thread  whenever  v^  -{-gz  becomes  negative. 


236  DYNAMICS. 


Of  the  Motion  of  a  material  Point  on  the  Arc  of  a  Cycloid. 

459.  Let  a  material  point  M  [Pig.  178)  be  supposed  to 
move  fram  rest  on  the  arc  of  a  cycloid,  under  the  influence  of 
the  force  of  gravity  :  the  initial  velocity  being  by  hypothesis 
equal  to  zero,  the  equation  (198)  is  reduced  to 

v2  =2gZj 


or 


whence  we  deduce 


dt=      ^ 


Let  the  origin  of  co-ordinates  be  assumed  at  the  point  E,  the 
absciss  ED  of  the  point  M'  being  denoted  by  w,  and  the  absciss 
EC  of  the  point  of  departure  by  A  :  we  shall  then  have 

CD=EC-ED; 
or, 

z—h — u. 

This  value  being  substituted  in  the  preceding  equation  givesi 
dt=^—--^^ (215). 

This  equation  contains  three  variables  ;  we  must  therefore 
eliminate  one  by  means  of  the  equation  of  the  cycloid.  For 
this  purpose,  let  2a  represent  the  diameter  BE  of  the  gene- 
rating circle,  and  x  and  o/  the  co-ordinates  AP  and  PM' 
of  the  point  M',  reckoned  from  A  as  an  origin  ;  the  equation 
of  the  curve  will  then  be 

"^^TW^^ ^^^«>- 

But  if  s  denote  the  arc  AM',  we  shall  have  the  relation 

ds=^{dx''  +dj/''); 
or, 


MOTION  UPON  A  CYCLOID.  237 

substituting  in  this  equation  the  value  of  -7-  deduced  from 
the  relation  (216),  we  find 

or, 

^='y\^^y (^^^- 

But  from  an  inspection  of  the  figure,  we  have 

2a — y=u  ; 
and  hence, 

dy= — du. 
By  substituting  these  values  in  (217),  we  obtain 

ds=^ — du\^/ — . 
V      u 

The  difierentials  of  5  and  u  have  contrary  signs,  since  the 

first  is  a  decreasing  function  of  the  second. 

The  preceding  value  of  ds  will  reduce  equation  (215)  to 

du 

g'  ^{hu—u^) 

460.  This  equation  may  be  integrated  by  the  formula 

dx 


dt=-y/^.      ,/^     ^, (218). 

lis  eq 

/y/o"^  ^3^=arc  (versed  sine  ~x)  ; 


for  by  making  x=—,  this  formula  becomes 

— 7x r=arc  I  versed  sine  =— | (219)  ; 

^{2az—z')  \  a  F  v       /> 

and  consequently,  by  referring  the  integral  of  equation  (21 8) 
to  this  formula,  we  obtain 

I^-k/--  .  arc  f  versed  sine=  ^\-\-C (220). 

^    g  \  \h} 

To  determine  the  constant,  we  remark,  that  since  the  time  is 

reckoned  from  the  instant  when  the  body  is  at  the  point  M, 

we  must  then  have 

/=0,  and  «=EC=A; 

this  supposition  reduces  the  equation  (220)  to 


238  DYNAMICS. 

0=  — \./  —  •  3-rc  (versed  sine  =2)4-C. 
^    g 

But  the  arc  whose  versed  sine  is  equal  to  2,  being  a  semi- 
circumference,  if  we  denote  by  v  the  semi-circumference  of  a 
circle  whose  radius  is  unity,  the  preceding  equation  will 
become 

This  value  will  reduce  equation  (220)  to 

i=>v/— fw— arc  (versed  sine  =-t-|. 

This  expression  gives  the  time  of  descent  to  the  point  M',  the 
absciss  of  which  is  equal  to  u.  To  obtain  the  entire  time 
of  descent  to  the  vertex  E,  we  make  w=0,  and  the  value  of 
t  is  then  reduced  to 


'-^\/~t 


g 

This  value  of  the  time  being  independent  of  the  height  h 
of  the  point  of  departure,  we  conclude  that  the  time  necessary 
for  a  material  point  to  deseend  to  the  vertex  E  of  the  cycloid, 
under  the  influence  of  the  force  of  gravity,  is  constantly  the 
same,  whatever  may  be  the  position  of  its  point  of  departure. 

Of  Oscillatory  Motion. 

461.  Let  OBC  {Fig.  179)  represent  a  continuous  curve, 
intersected  at  the  points  O  and  C  by  a  horizontal  line,  and 
supposed  to  contain  no  angular  points  that  might  occasion  a 
loss  of  velocity  to  a  body  or  material  point  moving  upon  it. 
Let  the  tangent  BT  at  the  point  B  be  supposed  horizontal, 
the  co-ordinate  plane  of  x,  y  being  likewise  horizontal.  If 
the  co-ordinates  z  be  reckoned  positive  downwards,  we 
shall  have  the  following  equations  to  determine  the  circum- 
stances of  the  motion  of  a  material  point  sliding  along  the 
curve  under  the  influence  of  gravity  : 


OSCILLATORY   MOTION.  239 

To  determine  the  velocity  of  the  moveable  point,  we  proceed 
as  in  Art.  433  :  multiplying  these  equations  by  2da:,  2dy,  and 
2dz  respectively,  and  adding,  we  find 
2dxd^  X + 2dyd^  y + 2dzd''  z 


2gdz', 


dt^ 
and  by  integration, 

——de ^^^+^' 

or, 

Replacing-—  by  its  value  v^,  there  will  result 

If  V  denote  the  velocity  at  the  point  O,  when  ;$r=Oj  the  pre- 
ceding equation  will  become 

and  consequently,  by  substituting  this  value  of  C,  we  shall 
obtain 

'V^=Y^+2gz (221). 

462.  Since  the  ordinates  increase  from  the  point  O  to  the 
point  B,  it  appears  from  equation  (221)  that  the  motion  will 
be  accelerated  while  the  material  point  is  describing  the  arc 
OB,  and  that  its  velocity  will  be  a  maximum  at  the  point  B  : 
the  ordinates  decreasing  beyond  this  point,  the  velocity  of  the 
moveable  point  will  likewise  be  diminished.  This  diminution 
must  be  such  that  the  material  point  will  have  at  the  point 
m!.  the  same  velocity  as  it  previously  had  at  the  point  m^ 
situated  in  the  same  horizontal  plane  ;  for  the  vertical  ordi- 
nates of  these  points  being  equal,  their  values  substituted  in 
equation  (221)  will  necessarily  give  the  same  values  for  the 
two  velocities. 

The  velocity  diminishing  constantly  with  the  arc  Om,  we 
shall  find  on  the  prolongation  of  this  arc  a  point  A  at  which 
this  velocity  will  have  been  equal  to  zero  ;  and  the  moveable 
point  may  therefore  be  considered  as  moving  from  rest  at  this 
point.  If  through  the  point  A  a  horizontal  line  AA'  be 
drawn,  intersecting  the  second  branch  of  the  curve  at  A',  the 


240  DYNAMICS. 

velocity  at  A'  will  likewise  be  equal  to  zero.  Thus,  the 
motion  will  cease  at  the  point  A',  and  the  action  of  gravity, 
causing  it  to  descend  from  A'  to  B,  will  augment  the  velocity 
in  the  same  manner  that  it  was  before  diminished.  At  the 
point  B  the  velocity  will  again  become  a  maximum,  and  the 
moveable  point  will  then  ascend  to  the  point  of  departure  A, 
its  motion  being  retarded  in  the  same  manner  that  it  was 
before  accelerated  in  descending  from  A  to  B. 

The  same  efiects  being  repeated  by  the  action  of  gravity, 
the  point  will  continue  to  oscillate  indefinitely. 

If  the  arcs  AB  and  A'B  are  similar,  the  times  of  describing 
them  will  evidently  be  equal.  When  the  oscillations  of  a 
body  or  material  point  are  all  performed  in  equal  times,  they 
are  said  to  be  isochroiial. 

463.  Let  B'OBO'  {Fig.  180)  represent  a  curve  returning 
into  itself,  and  symmetrical  with  respect  to  a  vertical  axis 
passing  through  the  points  B  and  B'  at  which  the  tangents  are 
horizontal.  If  the  material  point  descend  from  a  point  O, 
with  an  initial  velocity  such  that  upon  arriving  at  B  it  can 
ascend  from  B  to  B'  on  the  second  branch  of  the  curve,  it  will 
descend  a  second  time  on  the  arc  B'OB,  the  force  of  gravity 
restoring  the  velocity  lost  during  the  ascent  on  the  arc  BO'B'. 
The  same  effects  being  repeated,  the  body  will  continue  to 
revolve  indefinitely. 


Of  the  Simple  Pendulum. 

\  464.  The  simple  pendulum  is  composed  of  a  material 
heavy  point  M  {Fig.  181),  suspended  by  an  inflexible  right 
line  MC  devoid  of  weight,  and  oscillating  about  a  point  C. 
In  this  motion  the  point  M  describes  the  arc  of  a  circle  about 
C  as  a  centre,  and  the  velocity  of  M  will  be  given  (Art.  444) 
by  the  equation 

v^  =  V2  -[■2gz (222). 

ds 
Replacing  v  by  its  value  —,  we  find 

dt=     ^^/\     , (223). 


SIMPLE    PENDULUM.  241 

The  origin  being  acsiimed  at  the  point  of  departure,  z  will 
represent  the  ordinate  M'P'  {Fig.  182)  of  the  point  M',  at 
which  the  material  point  is  found  after  the  lapse  of  a  certain 
time,  and  V*  will  represent  the  square  of  the  velocity  which 
the  body  has  at  the  point  M,  where  xr=0.  If  h  denote  the 
height  due  to  this  velocity,  we  shall  have  the  relation 

and  the  equations  (222)  and  (223)  therefore  become 

-v/[2=^(A+.)],    *=;7i^$+;)] (224) 

465.  To  express  the  quantity  z  in  functiors  of  the  co-ordi- 
nates of  the  circle  described  with  the  radius  CM,  we  demit 
the  perpendiculars  MB  and  M'D  on  the  vertical  line  CE,  and 
denote  by  a  the  radius  CE,  by  h  the  vertical  distance  EB, 
and  by  x  the  absciss  ED  of  the  point  M'  referred  to  the  point 
E  as  an  origin  ;  we  shall  then  have 

-  =  BD  =  6— .r. 

And  by  introducing  this  value  into  equations  (224),  they  be- 
come 

.=^[2^(,  +  6-.)],     d.=_^^^*__^ (223). 

From  the  first  of  these  relations  we  obtain  the  velocity  of  the 
material  point  at  the  point  M',  corresponding  to  the  absciss 
X  ;  the  second,  being  integrated,  will  determine  the  time  em- 
ployed by  M  in  descending  to  the  point  M',  To  effect  this 
integration,  we  must  eliminate  one  of  the  variables  contained 
in  the  second  member,  by  m-eans  of  the  relations 

ds=^iclx^-\-dij^) (226), 

y^=2ax—x^ (227). 

The  latter  being  differentiated,  gives 
ydy={a — x)dx  ; 
and  consequently, 

dy^=i^^IIpldx'. 

This  value  being  substituted  in  equation  (226),  we  find 

a  21 


242  DYNAMICS. 

or,  replacing  y^  by  its  value  given  in  equation  (227),  and  re- 
ducing, we  obtain 

,       ,         /a^       ,  adx      ,  adx  .  adx 

as=dx^  /  —  =  ± =  ± — -r r==t — ft^ r— t  5 

V  y2  y  ^(2ax—x')         ^[(2a—x)x]^ 

whence, 

~~  ^[{2a-x)x]^[2ff{hi-b-x)]' 

The  negative  sign  is  here  prefixed  to  the  second  member,  be- 
cause the  co-ordinate  a;  is  a  decreasing  function  of  the  time  t. 
466.  If  the  initial  velocity  be  supposed  equal  to  zero,  we 
shall  have 

h=0] 

and  if  at  the  same  time  the  arc  through  which  the  oscillation 
is  performed  be  supposed  extremely  small,  we  can  neglect  x 
in  comparison  with  2a,  and  the  value  of  dt  will  be  then  re- 
duced to 

,  adx 

^~^{2ax)^[2g{b~x)]- 

This  equation  may  be  put  under  the  form 

dt=—l\/-  X     .^f""    ,  , (228). 

The  value  of  i  will  be  immediately  obtained  by  an  integration 
of  the  formula 

dx 
:^{bx—x^) 
which,  by  a  comparison  with  (219),  gives 

a=^b,     z=x: 
and  by  substituting  these  values  in  equation  (219),  which  is 


f:77^) (^«>^ 


/ r-  =arc  I  versed  sine = —  | , 

J^(2az-z'')  \  a/' 


^{2az-z'') 
it  becomes 


/! :.^£- — -=arc  (versed  sine =^  | 
^{bx-x')  \  \b} 

=arc  (versed  sine=-^  j . 
But,  in  general,  the  cosine  of  the  arc  corresponding  to  the 


SIMPLE    PENDULUM.  243 

versed  sine  c  and  radius  unity,  being  equal  to  1 — c,  we  shall 
have 


arc 


(versed sine =—  j=arc  (cos=l — —\ 

)• 


/  b 

=arc|  cos=  — 


V  b 

This  value  of  the  expression  (229)  being  substituted  in  equa- 
tion (228),  we  shall  find 

^=-i\/-Xarc(cos=^::^Wc (230). 

467.  The  constant  may  be  determined  from  the  consider- 
ation that  when  t=0,  x=b  ',  these  values  reduce  the  equation 
(230)  to 

0=-!^/- Xarc(cos=_l)-f  C. 

If  «•  denote  the  semi-circumference  of  a  circle  whose  radius 
is  unity,  we  shall  have  (Pig:  174) 

arc  (cos = — 1)  =  ar c  BC A = sr  ; 
and  consequently. 

By  substituting  this  value  in  the  equation  (230),  we  obtain 

The  integral  being  taken  between  the  limits  ar=6,  which  cor- 
responds to  ^=0,  and  x  equal  to  any  assumed  absciss,  will 
make  known  the  time  of  descent  from  the  point  M  {Fig:  182) 
to  the  point  M'  corresponding  to  the  assumed  value  of  x. 

468.  When  we  wish  to  obtain  the  time  of  descent  to  the 
lowest  point  E,  we  make  x=0,  in  the  preceding  expression  ; 
and  since  the  arc  whose  cosine  is  unity  is  equal  to  zero,  we 
shall  have 

<=iTv/-^ (232). 

^   g- 

469.  When  the  material  point  arrives  at  the  point  E,  it 

Gt2 


244  DYNAMICS. 

will  have  acquired  its  maximum  velocity  ;  for  the  velocity 
being  expressed  by 

it  will  evidently  be  a  maximum  at  that  point  of  which  the 
ordinate  z  is  the  greatest.  Thus,  in  virtue  of  the  velocity 
acquired  at  E,  the  moveable  point  will  describe  the  arc  EN  ; 
and  since  this  arc  changes  its  sign  in  passing  through  zero, 
we  find  for  the  expression  of  the  time  requisite  for  the  point 
to  arrive  at  N' 

^=i\/|[-+arc(cos  =1-^)] (233). 

If  from  this  expression  we  subtract  that  given  by  (232), 
which  expresses  the  time  of  descent  from  the  point  M  to  the 
point  E,  there  will  remain 

I S/—  X  arc  / cos=  1  — — V 

an  expression  Ibr  the  time  of  ascent  from  E  to  N':  this  time 
is  equal  to  that  employed  in  descending  from  M'  to  E,  as  may 
be  proved  by  taking  the  difference  between  equations  (231) 
and  (232). 

Finally,  when  the  material  point  shall  have  arrived  at  the 
point  Nj  situated  in  the  horizontal  line  passing  through  M,  we 

shall  ]iave  j:— 6,  and  the  expression  arc  |cos=l — —I  will 

then  become  arc  (cos  =— l)=5r;  thus,  the  equation  (233) 
will  be  reduced  to 

^   g 
Such  will  be  the  value  of  the  time  required  by  the  moveable 
point  to  describe  the  whole  arc  MEN.     This  time  being  de- 
noted by  T,  we  have 

T='^\/- (234). 

The  velocity  of  the  material  point  upon  its  arrival  at  the  point 
N  will  be  equal  to  zero  ;  for,  since  the  initial  velocity  was 
supposed  equal  to  zero,  we  have 

A=0; 


SIMPLE    PENDULUM.  245 

and  this  value,  taken  in  connexion  with  that  of  x=b,  reduces 
the  equation  v=^[2g{k-^b—x)]  to 

v=0. 
The  motion  of  the  material  point  being  entirely  destroyed 
when  it  arrives  at  the  point  N,  the  force  of  gravity  will  cause 
it  again  to  descend,  and  since  the  circumstances  of  the 
motion  are  precisely  similar  to  those  presented  when  the 
point  commenced  its  motion  at  M,  a  second  oscillation  will 
be  performed  in  the  same  time,  and  a  similar  motion  will 
continue  indefinitely. 

470.  The  equation  (234)  being  independent  of  the  quantity 
b  which  expresses  the  vertical  distance  MK,  it  follows  that 
if  the  point  of  departure  had  been  taken  at  M',  instead  of  at 
M,  the  time  of  oscillation  would  have  been  the  same  ;  and 
consequently,  that  if  several  material  points  depart  from  the 
different  points  M,  M',  M",  (fcc,  they  will  all  perform  their 
oscillations  in  the  same  time.  It  should  be  recollected,  how- 
ever, that  this  result  has  only  been  obtained  on  the  supposition 
that  the  arcs  described  are  extremely  small. 

471.  These  oscillations  of  equal  duration  are  called  iso- 
chrmial.  But  if  the  length  of  the  pendulum  be  supposed  varia- 
ble, the  time  of  vibration  will  likewise  vary  :  for,  if  I  and  V 
represent  the  lengths  of  two  pendulums,  whose  oscillations 
are  performed  in  the  times  T  and  T',  we  shall  have 

g  ^    g 

hence, 

T  :  T'  :  :  v^Z  :  v^Z' (235). 

Thus,  if  the  time  of  oscillation  T  of  one  pendulum  be  accu- 
rately known,  we  can  determine  by  the  preceding  proportion 
the  length  V  of  a  pendulum  which  shall  vibrate  in  an  arbi- 
trary time  T'. 

472.  To  ascertain  with  greater  precision  the  time  of  a 
single  oscillation,  we  will  represent  by  N  the  number  of  oscil- 
lations made  by  the  pendulum  whose  length  is  I  in  a  time  6, 
and  by  N'  the  number  of  oscillations  of  the  pendulum  I'  in 
the  same  time  «  :  we  shall  then  have 


t=Vf' '^-Vl' 


246  DYNAMICS. 

T=^,    andT'=^, (236). 

By  means  of  these  values,  the  proportion  (235)  is  reduced  to 

N'^  :  N=  :  :  ^  :  l\ 
whence, 

When  the  number  of  oscillations  made  by  a  pendulum  of  a 
given  length,  in  a  given  time,  has  been  ascertained  from 
observation,  we  can  calculate  the  length  of  the  pendulum 
which  will  oscillate  in  a  second  of  time. 

If  an  error  be  committed  in  observing  the  time  Ô,  this  error 
will  he  greatly  reduced  by  being  divided  by  the  number  of 
Dscillations,  and  if  this  number  be  taken  large,  the  effect  of 
the  error  upon  the  time  of  a  single  vibration  may  be  regarded 
as  insensible. 

473.  It  is  jXi  this  principle  that  the  length  of  the  seconds 
pendulum,  which  makes  86,400  oscillations  in  a  mean  solar 
day,in  vacuo,  and  at  the  latitude  of  New- York,  has  been  found 
■equal  to 

in.  ft. 

39.10168=3.25847,  nearly. 

474.  To  determine  the  value  of  g,  the  measure  of  the  in- 
tensity of  the  force  of  gravity,  we  employ  the  equation  (234)j 
which  gives 

and  by  making  in  this  equation 

in.  ft. 

T=l",     Z=39.1G168,     and  «-=3.1415926, 

or,. 

a-2  =9.8696046; 
sire  find 

in.  ft. 

^=385.9183=32.1598. 

475.  If  g  and  g'  represent  the  intensities  of  gravity  at  dif- 
ferent places,  and  I  and  V  the  lengths  of  two  pendulums 
which  oscillate  in  the  times  T  and  T',  we  shall  have 


T=-V'^-,    T^'s/!:' 


g'  ^  g" 


CENTRIFUGAL    FORCE.  247 

ffom  which  we  deduce 

T:T'::  v/|:  V^l (237)- 

Let  N  and  N'  represent  the  numbers  of  oscillations  made  by 
these  pendulums  in  the  time  6  ;  T  and  T'  will  be  given  in 
functions  of  6  by  equations  (236),  and  their  values  being  sub- 
stituted in  the  proportion  (237),  will  give,  after  reduction^ 

If  the  same  pendulum  be  used  at  the  two  places,  Z  and  V  will 
be  equal  to  each  other,  and  the  preceding  proportion  will 
become 


whence,, 


1  -1  ••      /I-      /L- 


N  ■  N'  ■  '  V  ^  ■  V  ^' 


N' 


Of  the  Centrifugal  Force. 

476.  If  a  material  point  be  supposed  to  move  around  a 
fixed  centre  C,  describing  the  curve  LMK  {Pig.  183),  and  if, 
upon  its  arrival  at  the  point  L,  the  connexion  with  the  centre 
be  suddenly  destroyed,  the  material  point  will,  in  virtue  of 
the  law  of  inertia,  continue  to  move  in  the  direction  of  the 
tangent  LT.  But  if  we  conceive  the  point  to  be  compelled 
to  describe  the  curve,  it  will  leave  the  tangent,  and  will  after 
a  certain  time  arrive  at  the  point  M.  The  arc  LM  being  sup- 
posed indefinitely  small,  the  angle  LCM  will  be  so  likewise, 
and  the  lines  LC  and  MC  may  be  considered  as  parallel. 
Thus,  replacing  CM  by  the  parallel  CM,  and  constructing 
the  parallelogram  LDMN,  it  appears  that  the  material  point, 
if  free,  would  describe  the  side  LD,  while  by  its  connexion, 
with  the  fixed  centre  it  is  caused  to  describe  the  diagonal 
LM  ;  the  effect  of  the  force  which  draws  the  point  towards  the 
centre  has  therefore  been  to  move  it  through  the  space  MD. 

The  point  may  be  supposed  to  be  retained  on  the  curve 
LMK,  either  in  virtue  of  a  force  of  attraction  which  is  con- 
stantly directed  towards  the  centre  C,  or  by  the  resistance 


248  DYNAMICS. 

opposed  by  the  curve  regarded  as  material  ;  or,  finally,  by 
being  connected  with  the  point  C,  by  means  of  a  cord  of 
variable  length. 

Whilst  the  point  is  describing  the  elementary  arc  LM,  we 
can  regard  it  as  moving  upon  the  equal  arc  of  the  osculatory 
circle,  and  can  suppose  it  to  be  retained  on  this  arc  by  means 
of  a  thread  of  an  invariable  length,  attached  to  the  centre  of 
the  osculatory  circle.  Moreover,  since  this  thread  will  ex- 
perience a  tension  only  in  consequence  of  the  resistance 
offered  by  the  material  point  to  the  force  which  tends  to 
deflect  it  from  the  tangent,  this  tension  or  the  resistance  op- 
posed by  the  point  will  be  precisely  equal  to  the  force  which 
causes  it  to  deviate  from  the  tangent.  This  resistance  is 
exerted  in  the  direction  of  the  radius  of  curvature,  and  its 
constant  tendency  is  to  remove  the  material  point  from  the 
centre  of  curvature.  Hence,  it  is  called  the  centrifugal 
force  ;  and  the  force  which  constantly  urges  a  body  towards 
any  fixed  centre  is  called  a  centj'ipetal  force. 

The  centrifugal  force  evidently  corresponds  to  the  quantity 

represented  by  —  in  Arts.  451  and  452. 

7 

477.  To  determine  directly  the  expression  for  the  centri- 
fugal force,  we  replace  the  infinitely  small  arc  LM  by  the 
chord  of  the  osculatory  circle  at  the  point  L  {Pig.  184). 
Then,  the  versed  sine  LN  will  represent  the  space  through 
which  the  point  would  be  drawn  in  virtue  of  the  action  of 
the  centrifugal  force,  during  the  time  occupied  by  the  point 
in  describing  the  arc  LM.  From  the  known  property  of  the 
circle,  we  have 

LN  :  LM  :  :  LM  :  LE  ; 

or,  by  substituting  the  arc  for  its  equal  the  chord, 
hence, 

and  by  substituting  for  ds  its  value  vdt,  we  find 
LN^"?^ (238). 

2y 


CENTRIFUGAL    FORCE.  249 

A  second  expression  for  the  value  of  LN  may  be  obtained  in 
the  following  manner.  The  time  required  to  describe  the 
arc  LM  will  be  represented  by  dt,  since  this  arc  is  itself 
denoted  by  ds  ;  hence,  dt  will  likewise  represent  the  time 
in  which  the  material  point  would  be  caused  to  describe  a 
space  equal  to  LN  under  the  influence  af  the  centrifugal 
force  alone.  Moreover,  the  centrifugal  force  acts  incessantly, 
and  during  the  infinitely  short  time  dt,  its  intensity  may  be 
considered  invariable.  If,  therefore,  we  regard  this  force  as 
constant,  and  denote  its  intensity  by/,  the  circumstances  of 
motion  of  the  point,  under  the  influence  of  this  force,  will  be 
expressed  by  the  equations 

dv „     ds  _ 

di~^'  ir'''' 

and  by  integration, 

But  the  space  LN  being  that  which  corresponds  to  the  time 
dt,  if  in  the  preceding  equations  we  make  LN=5^  t  will  be- 
come dt  ;  we  shall  thus  have 

LN  =  ^c?PX/. 
This  value  of  LN  being  substituted  in  equation  (238)  gives, 
after  reduction, 

y 

478.  If  the  material  point  be  supposed  to  have  a  circular 
motion, — as,  for  example,  when  a  stone  is  whirled  round  in 
a  sling,  V  will  become  the  radius  of  the  circle  described,,  and 
the  expression  for  the  centrifugal  force  will  then  be 

/=^ (239). 

Let  h  represent  the  height  due  to  the  velocity  v  ;  the  follow 
ing  relation  will  then  subsist  (Art.  401), 

eliminating  v^  between  this  equation  and  that  which  pre- 
cedes,  we  obtain 

f^2h 


250  DYNAMICS. 

from  which  we  conclude,  that  the  centrifugal  force  is  to  the 
force  of  gravity  J  as  twice  the  height  due  to  the  velocity  is  to 
the  radius  of  the  circle  described  by  the  material  point. 

479,  If  a  semicircle  EAF  {Pig.  185)  be  supposed  to  re- 
volve about  its  diameter  EF=2R,  the  point  A,  the  middle  of 
arc  EAF, will  describe  a  circumference  equal  to  25rR  ;  if  this 
motion  be  performed  uniformly  in  the  time  T,  with  the  ve- 
locity V,  we  shall  have  the  relation 

vxT=2^R; 
and  by  eliminating  v  between  this  equation  and  (239),  we 
find 

/=^ (240). 

In  like  manner,  if/'  represent  the  centrifugal  force  of  a  point 
which  describes  uniformly  the  circumference  of  a  circle 
whose  radius  is  R',  in  the  time  T',  we  shall  have 


T' 


and  consequently. 


From  this  proportion  we  immediately  conclude,  that  when  the 
radii  R  atid  R'  are  equal,  the  centrifugal  forces  will  he  in  the 
inverse  ratio  of  the  squares  of  the  times  of  revolution  ;  and 
that  when  the  times  are  equal,  the  forces  will  be  directly  as 
the  radii. 

480.  The  effect  of  the  centrifugal  force  at  the  equator, 
caused  by  the  revolution  of  the  earth  upon  its  axis,  can  now 
be  estimated.  For,  the  equatorial  radius  of  the  earth  being 
20920300  feet,  we  replace  R  by  this  value  in  equation  (240), 
substituting  at  the  same  time  the  values  of  sr  and  T.  But 
we  have,  approximatively, 

«■=3.1415926,     ^='=9.8696046. 

The  time  T  is  determined  from  the  consideration  that  the 

earth  performs  a  revolution  upon  its  axis  in  0.997269  days, 

the  day  being  composed  of  86400  seconds.  Thus  we  shall  have 

T=0.997269  x  86400"  ==86 164". 


CENTRIFUGAL    FORCE.  251 

Substituting  this  value  and  that  of  R  in  equation  (240),  there 
results 

/=0?Ï112 (242). 

481.  Having  found  the  value  of  /,  we  can  determine  the 
intensity  G  of  the  force  of  gravity  which  would  be  observed 
at  the  equator  if  the  earth  were  immoveable.  For,  since  the 
force  /  is  directly  opposed  to  the  force  G,  a  portion  of  the 
latter  will  be  destroyed  by/;  and  hence,  if  g  denote  the  in- 
tensity of  gravity  as  determined  by  observation,  we  shall 
have 

^=G-/; 
or, 

substituting  in  this  expression  the  value  of/ given  by  equa- 
tion (242),  and  that  of  ^,  which  at  the  equator  is  32.0861  ft., 
we  find 

G-32.086H- Oil  12=32.1973 (243). 

To  determine  the  relation  between  the  centrifugal  force  and 
the  force  of  gravity,  we  divide  equation  (242)  by  equation 
(243),  which  gives 

/      0.1112"-      1            ,  ,„,,^ 

^=321973^-^2-89  "^"^^^ ^^^^^' 

482.  The  proportion  (241)  will  furnish  a  solution  to  the 
following  problem  : 

To  find  the  time  in  which  a  revolution  of  the  earth  should 
be  performed,  in  order  that  the  centrifugal  force  at  the  equa- 
tor may  he  equal  to  the  force  of  gravity. 

Let  T'  represent  the  required  time  of  revolution,  and/'  the 
corresponding  centrifugal  force  ;  we  shall  then  have,  by  the 
nature  of  the  problem, 

/'=G,  and  R'=R  ; 
these  values  substituted  in  the  proportion  (241)  reduce  it  to 

*  *    r£â    '  fjvâ  • 

whence  we  obtain 


252 


DYNAMICS. 


f 

If  the  fraction  --  be  now  replaced  by  its  value  (244),  we  shall 
G 

find 

T        T 

v/2S9     17' 
Thus,  if  the  earths  rotation  were  seventeen  tinier  more  rapid 
tliati  it  actually  is,  the  centrifugal  force  at  the  equator  would 
be  equal  to  the  gravity. 

483.  To  find  the  diminution  of  the  gravity  produced  by 
the  centrifugal  force  at  any  other  point  on  the  earth's  surface, 
it  will  be  necessary  to  determine  the  effect  of  the  centrifugal 
force  in  the  direction  of  the  vertical  BZ  {Fig.  185)  drawn 
through  the  point  under  consideration. 

For  this  purpose,  we  will  regard  the  earth  as  spherical,  it 
being  nearly  so  :  the  latitude  of  the  point  B  being  then  repre- 
sented by  the  arc  AB,  it  will  be  measured  by  the  angle 

BOA=ZBC=^^. 
Denoting  by  R  the  radius  AO  of  the  earth,  and  by  R'  the 
radius  BD  of  the  parallel  of  latitude  passing  through  B,  we 
shall  have 

R'=R  cos  OBD  ; 
or, 

R'=R  cos  ^. 

Let  the  centrifugal  force  at  the  point  B,  which  is  exerted  in 
the  direction  of  the  radius  DB,  be  represented  by  the  line  BC, 
and  resolve  it  into  the  two  components  B6  and  Be.     The 

force  BC  will,  by  Art.  479,  be  expressed  by  -^- — ,  and  the 

component  /'  in  the  direction  of  the  vertical  BZ,  which  is 
represented  by  B6,  \vill  be  given  by  the  relation 

/=->p,-Xcos^: 

and  by  substituting  in  this  relation  the  value  of  R',  we  shall 
obtain 

/--Tj^XCOS^'r/.. 

4     3f? 

The  factor   -=^  represents  the  centrifugal  force  /  at  the 


GRAVITATION.  253 

equator  ;  this  equation  may  therefore  be  transformed  into  tlie 
proportion 

/:/::  1  :  cos-  ^] 
from  which  we  conchide,  that  the  dimhiations  of  gravity  at 
different  places  on  the  earth's  surface,  arising  from  the  action 
of  the  cetitrifu  gal  force,  are  proportional  to  the  squares  of  the 
cosines  of  the  latitudes. 

484.  The  latitude  of  New-York  beinor  40°  42' 40",  its 
cosine  will  be 

0.7580  ; 
and  by  multiplying-  the  value  of/ (242)  by  the  square  of  this 
number,  or  by 

0.5746, 
we  find 

/=o!b639. 
If  G'  represent  the  value  of  the  force  of  attraction,  or  that 
which  the  observed  gravity  would  have  in  the  latitude  of 
New- York,  if  the  earth  were  immoveable,  the  gravity  actually 
observed  being  denoted  by  g',  we  shall  ha-'e,  as  in  Art.  481, 

G'=g'+f. 
The  observed  gravity  g',  in  the  latitude  of  New- York,  being 

32!l598, 
we  find,  by  substituting  this  value  and  that  of/'  in  the  preced- 
ing equation, 

G'=32?i598+o'.0639=32!2237 (245). 

Of  the  Systeiyi  of  the  World. 

485.  In  discussing  the  properties  of  the  centre  of  gravity, 
we  have  already  had  occasion  to  consider  that  remarkable 
force  exerted  by  the  earth,  in  virtue  of  which  all  bodies  are 
solicited  in  directions  perpendicular  to  its  surface.  The  ex- 
istence of  this  force  was  not  entirely  unknown  to  the  ancients  : 
Anaxagoras,  and  his  disciples,  Democritus,  Plutarch,  Epi- 
curus, and  others,  admitted  the  existence  of  such  a  force  ;  and 
similar  opinions  were  entertained  by  Kepler,  Galileo,  Huy- 
gens,  Fermât,  Roberval,  <fec.,  in  modern  times.    The  celebrated 

22 


254 


DYNAMICS. 


Kepler  distinctly  affirms,  in  his  work  De  Stella  Martis,  that 
the  force  of  attraction  is  not  confined  to  bodies  situated  upon 
the  surface  of  the  earth,  but  that  it  extends  to  the  most  distant 
stars. 

This  bold  conception  remained  \on^  unimproved,  from  the 
difficulty  of  verifying  its  truth.  The  eifects  of  gravity  at  the 
earth's  surface  were  measured  by  Galileo. 

Lord  Bacon,  suspecting  that  the  intensity  of  this  force  must 
vary  with  the  distance  from  the  centre  of  the  earth,  endeav- 
oured to  verify  the  truth  of  this  conjecture  by  observing  the 
distances  through  which  bodies  would  fall,  in  a  given  time, 
at  different  elevations  above  the  surface  of  the  earth.  But, 
however  great  were  these  elevations,  they  proved  too  small  to 
render  the  variations  in  the  intensity  of  gravity  perceptible. 

Newton  extended  his  views  yet  further  ;  and  not  satisfied 
with  the  mere  conjecture  that  the  intensity  of  gravity  was 
subject  to  variation,  he  endeavoured  to  measure  the  law  of  its 
diminution.  He  adopted,  as  the  most  probable  law  of  diminu- 
tion, that  of  the  inverse  ratio  of  the  square  of  the  distance  ; 
such  being  the  law  according  to  which  light  and  other  emana- 
tions were  known  to  be  propagated.  To  test  the  truth  of  this 
supposition,  he  endeavoured  to  obtain  a  measure  of  the  inten- 
sity of  gravity  at  the  distance  of  the  moon,  and  the  only 
obstacle  to  this  determination  arose  from  an  imperfect  know- 
ledge of  the  moon's  distance,  and  of  the  dimensions  of  the 
earth  ;  but  more  exact  determinations  of  these  elements  having 
been  supplied  by  Picard  and  others,  he  was  enabled  to  base 
his  calculations  on  more  accurate  data. 

486.  The  first  element  to  be  determined  in  this  investi- 
gation, is  the  intensity  of  gravity  at  the  surface  of  the  earth. 
The  method  of  obtaining  this  quantity  by  the  oscillations  of 
a  pendulum  has  already  been  explained  in  Arts.  474  and 
484  :  it  was  thus  found,  that  in  the  latitude  of  New- York, 
and  on  the  supposition  that  the  earth  was  immoveable, 

G'=32.2237 (246). 

This  quantity  is  nearly  the  same  for  all  places  on  the  surface 
of  the  earth. 

To  ascertain  the  diminution  which  the  intensity  of  gravity 
should  sustain  at  the  distance  of  the  moon,  according  to  the 


GRAVITATION.  255 

supposed  law  of  Newton,  it  will  be  necessary  to  know  the 
distance  of  the  moon  from  the  centre  of  the  earth.  This  dis- 
tance depends  on  the  horizontal  parallax  of  the  moon. 

487.  Let  CL  and  HL  {^Pig.  186)  represent  two  lines  drawn 
from  the  moon  to  the  two  extremities  of  the  terrestrial  radius, 
the  line  HL  being  perpendicular  to  this  radius.  The  angle 
HLC  is  called  the  horizontal  parallax  of  the  moon,  and  its 
mean  value,  according  to  Delambre,  is  57'.  If  therefore,  the 
radius  of  the  earth  be  taken  as  unity,  we  shall  have 

CLsinL=CH  =  l, 
and  consequently, 

CL=-^— =60.314; 
sin  57' 

this  value  differs  but  little  from  that  employed  by  Newton, 
who  supposed  the  mean  distance  to  be  60|. 

488.  If  we  denote  by  y  the  intensity  of  gravity  at  the  dis- 
tance of  the  moon,  upon  the  hypothesis  that  it  decreases  ac- 
cording to  the  law  of  the  inverse  ratio  of  the  square  of  the 
distance,  and  by  s'  the  space  which  it  would  cause  a  body  to 
describe  in  the  time  t^  we  shall  have 

G':y  ::  (60.314)3  :  1»  ; 

whence, 

_       ^' 

^""  (60.314)»' 

Such  is  the  expression  for  the  velocity  which  should  be  im- 
parted by  gravity,  at  the  distance  of  the  moon,  in  a  second 
of  time,  if  the  hypothesis  assumed  be  correct. 

489.  By  substituting  this  value  for  g  in  the  general  formula, 

s^\gt\ 
and  replacing  5  by  ^',  we  shall  obtain  the  space  described  in 
the  time  t. 

Thus,  if  we  suppose  the  time  to  be  one  minute,  or  60",  we 
shall  have  for  the  space  s\  which  the  body  would  describe  in 
a  minute  of  time, 

K.60)^' (247). 

*      (60.314)3  ^'^'^'f' 


256  DYNAMICS. 

490.  If  we  nefflect  the  decimal  fraction,  in  the  denomina- 
tor,  the  equation  reduces  to 

from  which  we  conclude  that  the  space  described  by  a  body- 
moving  from  rest,  in  a  minute  of  time,  at  the  distance  of  the 
moon,  should  be  equal,  according  to  Newton's  hypothesis,  to 
the  space  passed  over  in  a  second  of  time,  at  the  surface  of  the 
earth. 

But  if  we  take  account  of  the  decimal  fraction,  the  equation 
(247)  will  give  by  reduction, 

5'=JG'x0.9896; 
and  by  substituting  the  value  of  G'  (246),  we  have 

5'= ix32!2237x  0.9896; 
or,  by  performing  the  multiplications  indicated, 

5' =  15.9443 (248). 

Such  would  be  the  distance  described  by  the  body  in  a 
minute  of  time,  at  the  distance  of  the  moon^  if  the  body 
were  supposed  to  move  from  a  state  of  rest. 

491.  Let  us  now  examine  whether  this  result  is  confirmed 
by  experience.  For  this  purpose,  let  the  moon  when  at  its 
mean  distance  be  supposed  to  describe  the  arc  LM  {Pig.  187) 
in  a  minute  of  time  :  if  the  lines  LQ,  and  Q,M  be  drawn 
respectively  parallel  to  the  sine  and  versed  sine  of  the  arc 
LM,  we  may  regard  LM  as  the  diagonal  of  a  parallelogram 
of  which  LQ.  and  LP  will  be  the  sides.  If  the  moon  were 
not  solicited  by  the  earth's  attraction,  it  would  describe  the 
tangent  LQ,,  and  if  solicited  by  this  attraction  solely,  it  would 
describe  the  line  LP  in  the  same  time  :  this  line  LP  will  there- 
fore serve  to  determine  the  inten??ity  of  the  earth's  attraction, 
and  it  will  evidently  be  equal  to  the  versed  sine  of  the  angle 
LCM. 

But  since  the  mean  radius  r  of  the  moon's  orbit  undergoes 
but  a  very  slight  variation  in  a  minute  of  time,  this  portion 
of  the  orbit  may  be  regarded  as  the  arc  of  a  circle  described 
with  the  radius  r  ;  and  since  the  moon,  when  at  its  mean  dis- 
tance, moves  with  nearly  its  mean  velocity,  we  shall  have,  by 


GRAVITATION.  257 

calling  T  the  time  of  a  sidereal   revolution,  or  the  time 

required  by  the  moon  to  return  to  the  same  point  of  the 

heavens, 

T  :  1  minute  :  :  360°  :  angle  LCM  ; 

whence, 

,    T^,,     360° 
angle  LCM—        -. 

This  time  of  revolution  being  known  from  observation  to  be 
27  days  7  hours  43  minutes,  or  39343',  we  will  replace  T  by 
this  value,  at  the  same  time  reducing  the  degrees  to  seconds, 
to  render  the  division  possible  ;  we  thus  obtain 

angle  LCM=~:=32''.94. 

492.  The  question  is  thus  reduced  to  finding  the  versed 
sine  of  an  arc  of  32".94,  in  a  circle  described  with  the  mean 
radius  of  the  lunar  orbit. 

To  effect  this,  let  the  perpendicular  CI  {Fig.  187)  be  drawn 
to  the  middle  of  the  chord  LM  :  the  right-angled  triangles 
LMP,  LCI,  having  the  common  angle  L,  will  be  similar,  and 
will  give  the  proportion 

LC:IL::LM:LP; 
or, 

LC:IL::2IL:LP: 
whence, 

LP=^ (249). 

Let  e  represent  the  angle  LCI  equal  to  |LCM,  and  r  the  mean 
radius  LC,  we  shall  have 

YL=r.  sin  6, 

and  the  equation  (249)  will  become 

LP=2r.sin2  tf; 
or,  by  substituting  the  value  of  the  angle  6, 

LP=2r.sin''16".47 (250). 

If  a  denote  the  mean  radius  of  the  earth,  the  mean  radius 
of  the  lunar  orbit  will  be  expressed  by 
r=(60.314)a. 
R 


258 


DYNAMICS. 


493.  But  the  mean  radius  of  the  earth  determined  by  the 
measurement  of  a  degree  upon  its  surface,  being  equal  to 

20886500  feet, 
we  shall  have,  by  substitution, 

LP=60.314  x2x  20886500  Xsin2(16".47), 
and  changing 

sine  whose  radius  is  unity  .  ^      tabular  sine 

^    mtO   — ; — , 

1  tabular  radius 

for  the  purpose  of  using  logarithms,  we  find 

Log  60.314    ---------     1.7804181 

Log  2    - 0.3010300 

Log  20886500     --------    7.3198657 

Log  sin3  (16".47),  or2.log  sin  (16".47)    11.8041388 

Corresponding  number =16.0492  ft.     -     1.2054526 

494.  It  thus  appears  that  the  moon  falls  towards  the  earth 
in  a  minute  of  time  a  distance  of  16.0492  ft.,  corresponding 
very  nearly  with  that  deduced  on  the  supposition  that  the 
intensity  of  gravity  varies  inversely  as  the  square  of  the  dis- 
tance from  the  centre  of  the  earth.  The  diiference  between 
the  two  results  amounts  only  to  about  0.15  ft.  in  tlie  space 
fallen  through  by  the  moon  in  a  minute  of  time,  and  will 
consequently  become  nearly  insensible  in  the  space  described 
in  one  second.  Moreover,  this  slight  difference  might  fairly 
have  been  anticipated,  since  mean  values  of  the  several  quan- 
tities which  enter  into  the  calculation  have  alone  been  em- 
ployed. 

495.  The  remarkable  accordance  exhibited  by  the  preced- 
ing calculation  between  the  results  of  theory  and  experience, 
justifies  us  in  concluding  that  the  force  of  gravity  exerts 
an  influence  at  the  distance  of  the  moon,  but  that  its  inten- 
sity is  less  than  at  the  surface  of  the  earth,  in  the  inverse 
ratio  of  the  squares  of  the  distances  from  the  centre  of  the 
earth.  The  truth  of  this  supposition  has  been  uniformly 
confirmed  by  experience  ;  astronomical  tables  calculated  upon 
the  hypothesis  of  Newton  assign  the  positions  of  the  celestial 
bodies  such  as  they  are  determined  by  direct  observation. 


GRAVITATION.  259 

without  presenting  a  single  exception.  This  hypothesis  may 
therefore  be  regarded  as  fully  established  by  experience. 
Those  general  truths  which  are  designated  tJie  laws  of  Kep- 
ler, and  which  have  been  repeatedly  verified  by  observation^ 
serve  to  establish  the  hypothesis  of  Newton  in  the  most  clear 
and  decisive  manner.  These  laws  may  be  enunciated  as 
follows  : 

1°.  The  planets  describe  ellipses,  having  the  centre  of  the 
sun  at  one  of  their  foci, 

2°.  The  areas  of  the  elliptical  sectors  described  by  the 
radius  vector  draw7i  from  the  planet  to  the  centre  of  the  sun 
are  constantly  j)7'oportional  to  the  times  of  description* 

3°.  The  squai'es  of  the  times  of  revolution  of  the  several 
planets  are  proportional  to  the  cubes  of  their  mean  distances 
from  the  sun. 

496.  The  first  of  these  laws,  as  will  be  demonstrated,  is  a 
particular  case  resulting  from  the  more  general  law  of  nature, 
which  requires  that  a  body  subjected  to  the  action  of  a  force 
which  varies  inversely  as  the  square  of  the  distance  from  a 
fixed  point,  should  necessarily  describe  a  conic  section. 

The  second  law  has  already  been  noticed  (Art.  435),  and 
subsists  in  general  for  every  body  which  is  constantly  at- 
tracted towards  a  fixed  point.     The  question  is  thus  reduced 

*  In  the  general  course  of  reasoning  which  is  here  applied  to  the  motions  of 
the  planets,  these  bodies  are  regarded  as  mere  material  points.  The  propriety 
of  making  this  supposition  will  not  fully  appear  until  after  we  have  discussed 
the  circumstances  of  motion  of  a  solid  body  whose  several  particles  are  acted 
upon  by  incessant  forces.  It  will  then  be  found  that  the  motion  of  the  centre 
of  gravity  of  such  a  body  will  be  precisely  the  same  as  though  the  mass  of  the 
body  were  concentrated  at  its  centre  of  gravity,  and  the  several  forces  appliffd 
directly  to  ihat  point.  Thus  the  case  will  be  reduced  to  that  of  the  motion  of 
a  material  point.  It  is,  however,  quite  obvious  that  this  hypothesis  cannot 
differ  much  from  the  truth  ;  for,  since  the  dimensions  of  the  planets  are  exceed- 
ingly minute  when  compared  with  their  distances  from  the  sun,  it  follows  that 
every  particle  in  the  planet  will  be  acted  upon  by  a  force  which  is  very  nearly 
equal  and  parallel  to  the  force  exerted  upon  that  particle  which  coincides  with 
the  centre  of  gravity  of  the  planet.  Thus,  the  particles,  being  acted  upon  by 
parallel  and  equal  forces,  will  have  the  same  motions  as  though  they  were  uncon- 
nected with  each  other;  and  the  reasoning  may  be  applied  to  any  one  of  these 
particles,  the  central  one,  for  example. 

R2 


260  DYNAMICS. 

to  proving  the  truth  of  the  first  and  third  of  Kepler's  laws, 
after  adopting  the  hypothesis  of  Newton. 

497.  Let  the  origin  of  co-ordinates  be  placed  at  the  centre 
of  attraction  (Fig:  188),  which  corresponds  to  the  centre  of 
the  sun,  for  the  planetary  system,  and  let  R  denote  the  value 
of  the  force  of  attraction  exerted  by  the  sun  upon  one  of  the 
planets,  and  r  the  radius  vector  drawn  to  the  planet. 

The  force  R  coinciding  in  direction  with  the  radius  vector 
mA,  if  we  represent  by  <p  the  angle  mAP,  which  the  radius 
vector  forms  with  the  axis  of  a;,  the  components  of  the  force 
R  in  the  directions  of  the  axes  will  be 

X=R  cos  (p,     Y=R  sin  ç. 
But  in  the  right-angled  triangle  AwP,  we  have 

AP     .V        .         mP    y 

cos<p— — r=— J     sm<z'= — -=^. 
mA     r  niA    r 

Thus,  the  components  X  and  Y  of  the  force  R  will  be  ex- 
pressed by 

X=R-,     Y=R^: 
r  r 

and  since  the  incessant  force  is  supposed  to  act  in  the  direc- 
tion from  m  towards  A,  it  will  tend  to  diminish  the  co-ordi- 
nates AP=:r,  and  Pm=y,  of  the  point  m  :  hence,  the  compo- 
nents of  the  incessant  force  should  be  affected  with  the 
negative  sign  (Art.  51)  ;  the  two  preceding  equations  will 
thus  become 

■   r  r 

or,  replacing  X  and  Y  by  their  values  given  in  equations 
(180),  we  obtain 

^-_R^    ^ R^         r25n 

498.  For  the  purpose  of  integrating  these  equations,  let  the 
first  be  multiplied  by  y,  and  the  second  by  x  :  taking  the  dif- 
ference of  the  products,  and  multiplying  by  dt,  there  will 
result 

d'x       d'y    „ 
^  dt         dt        ' 


GRAVITATION.  261 


the  integral  of  which  is 

ydx — xdy  _ 


(252), 


dt 

the  arbitrary  constant  introduced  by  integration  being  denoted 
by  a. 

499.  To  obtain  a  second  integral,  we  multiply  the  first  of 
equations  (251)  by  2dx^  and  the  second  by  2c/y,  and  take  their 
sum  ;  we  thus  obtain, 

2dx  .  d^x-\-2dy  •  d'^y _     p„  Jxdx-\-ydy\  /okq\ 

■  11^  \        r        / ^      '' 

The  second  member  of  this  equation  containing  the  three 
variables  x,  y,  and  r,  we  eliminate  two  of  them  by  means  of 
the  relation  x^  -\-y^  —r'^ ,  which  gives,  by  differentiation, 

xdx-\-ydy=rdr  ; 
this  value  substituted  in  the  second  member  of  equation  (253) 
reduces  it  to 

2dx .  d'x-\-2dy  .  d'^y__nnr., 
df'  "~        '' 

or,  since  dt  is  regarded  as  constant, 

Integrating,  and  denoting  by  b  the  arbitrary  constant,  we  ob- 
tain 

^^l^^=b-2fRdr (254). 

500.  The  quantity  Rdr  is  affected  with  the  sign  of  integra- 
tion, the  intensity  of  the  force  R  being  supposed  a  function  of 
the  distance  r  ;  the  nature  of  this  function  will  remain  arbi- 
trary, so  long  as  we  do  not  adopt  a  particular  hypothesis. 

501.  This  equation  still  containing  three  variables,  we  re- 
duce the  number  to  two  {<p  and  r),  by  introducing  the  values 
ofx  and  y,  expressed  in  functions  of  r,  and  the  angle  ç  included 
between  the  radius  vector  and  the  axis  of  x  ;  these  values 
are  given  by  the  formulas, 

x=r  .  cos  <p,    y=r  .  sin  ^ (255). 

By  differentiating,  we  have, 

dx  =  —r  sin  (pd(p-{-cos  çdr  )  /qka\  . 

.    dy=     r  cos  ^c?^+sin  <pdr  S ' 


262  DYNAMICS. 

and  the  values  of  x,  y,  dx,  and  dy  given  by  equations  (255) 
and  (256),  being  substituted  in  equation  (252),  transform  it 
into 

-^"|=« (257). 

The  sum  of  the  squares  of  equations  (256)  gives,  after  re- 
duction, 

dx''-\-dy''=r^dç>''-\-dr^ (258) 

and  by  substituting  this  value  in  equation  (254),  we  obtain, 

'l^^^Èl=h-2f^dr (259). 

502.  To  determine  the  equation  of  the  curve  described  by 

the  moveable  point,  we  eliminate  dt  between  equations  (257) 

and  (259)  :  the  first  of  these  gives 

-  r-dçi 

dt  = ; 

a 

this  value,  introduced  into  the  second,  transforms  it  into 
a^r^dç)'^  +a^dr 
r*d<p^ 
whence  we  deduce, 

'^^r^{br^—a=  -2r-'fRdr) ^       ^* 

This  equation  being  integrated,  and  the  values  of  the  con- 
stants being  determined,  we  shall  have  a  relation  between  the 
radius  vector  r  and  the  angle  ^. 

503.  To  determine  the  constants  a  and  b,  we  will  resume 
the  integrals, 

ydx—xdy            r^dç>- -^-dr"^      ,      o/»t)j  /oai\ 

—dt"'     S3— =''-2-^^'- (^•'l'- 

The  integral  of  the  first  of  these  equations  is,  by  Art.  435, 

2  .  sector  LAm=a^ (262)  ; 

consequently,  by  making  ^=1,  we  shall  find  that  a  is  equal  to 
twice  the  sector  described  in  a  unit  of  time. 

The  same  result  may  be  obtained  from  the  equation 

r'f^^a (263); 

for  d/p  being  the  infinitely  small  arc  described  in  the  time  dt^ 


=  6— 2/Rrfr; 


GRAVITATION.  263 

by  a  point  on  the  radius  vector  whose  distance  from  the  centre 
of  attraction  is  equal  to  unity,  rd(p  will  be  the  arc  described  in 

the  same  time  with  the  radius  r  ;  hence  \r  .  rd<p,  or  —~-,  will 

represent  the  infinitely  small  sector  described  by  the  radius 
vector  in  the  time  dt  ;  but  since  the  areas  described  are  propor- 
tional to  the  times  of  description  (Art.  435),  we  can  find  the 
area  described  in  the  time  1,  by  the  proportion 

—^  :  dt  ::  area  described  in  time  unity  :  1  ; 
lit 

whence, 

area  described  in  time  unity  =  — -^  ; 

'      Idt  ' 

r"^  d0 
and  consequently,  —r-)  or  its  equal  a,  will  be  double  the  area 

described  by  the  radius  vector  in  the  unit  of  time. 

It  may  be  remarked  that  the  change  in  the  sign  of  the  first 
member  of  equation  (257)  which  converts  it  into  (263),  is 
merely  equivalent  to  a  change  in  the  position  of  the  fixed  line 
from  which  the  areas  described  are  reckoned  :  in  the  first 
case  they  are  reckoned  from  the  axis  of  y,  and  in  the  second 
from  the  axis  of  x. 

From  the  equation  (263)  we  deduce 

^==^ (264.) 

dt    r^  ^      ' 

The  quantity  —  expresses  the  angular  velocity  of  the  body, 

or  the  velocity  of  that  point  on  the  radius  vector  which  is  at 
the  distance  unity  from  the  centre  of  attraction  ;  and  it  ap- 
pears by  the  preceding  relation,  that  the  angular  velocity 
varies  in  the  inverse  ratio  of  the  square  of  the  radius  vector. 
504.  From  the  first  of  equations  (261),  we  may  infer  that 
the  quantity  a  is  independent  of  the  law  according  to  which 
the  attractive  force  is  supposed  to  vary  ;  but  the  quantity  b, 
which  appears  in  the  second  equation,  will  evidently  depend 
on  the  attractive  force,  which  likewise  appears  in  the  same 
equation.  It  will  therefore  be  necessary  to  adopt  some 
hypothesis  respecting  the  law  of  this  force,  such,  for  example, 


264  DYNAMICS. 

as  the  law  of  Newton,  which  supposes  that  different  bodies 
attract  each  other  in  the  direct  ratio  of  their  masses,  and  the 
inverse  ratio  of  the  squares  of  their  distances. 

Let  the  force  exerted  by  the  unit  of  mass,  at  the  distance 
>t,  be  denoted  by  1  ;  the  force  exerted  by  the  sun  upon  a 
body  placed  at  the  distance  k  will  then  be  expressed  by  the 
mass  M  of  the  sun,  or,  in  other  words,  by  the  number  of  units 
which  its  mass  contains  :  but  the  mass  of  a  planet  attracted 
by  the  sun  being  denoted  by  m,  this  planet  will  exert  an 
attraction  upon  the  sun,  which  will,  for  a  similar  reason,  be 
expressed  by  its  mass  m  :  moreover,  since  the  two  forces 
M  and  m  tend  to  cause  the  approach  of  the  two  bodies,  their 
effect  upon  the  relative  motion  of  the  bodies  will  be  the  same 
as  if  the  force  M  +  w  were  concentrated  in  the  sun,  and  acted 
on  the  planet  at  the  distance  k.  When  this  distance  varies 
and  becomes  equal  to  r,  the  intensity  of  the  force  will  like- 
wise vary.  Let  R  denote  its  intensity  at  the  distance  r  ;  the 
assumed  hypothesis  will  give  the  proportion 

M+7;ï:R::l:i; 
le-    r^ 

whence, 

R=^!1M+^) ^2g5)_ 

Such  is  the  value  of  the  attractive  force  which,  acting  at  the 
distance  r,  will  cause  the  bodies  to  approach  each  other. 

505.  The  value  thus  determined  corresponds  to  that  of  the 
incessant  force  which  we  have  hitherto  represented  by  R  : 
we  therefore  have 


/RA.=yiii(M&. 


putting,  for  brevity, 

/v2(M+m)=M' (266), 

the  preceding  equation  will  be  reduced  to 

'Wdr 


fRdr=fl 


■  ■  (267)  ; 

but  since  the  quantities  M  and  m  and  the  distance  k  remain 
invariable,  the  quantity  M'  will  be  constant  ;  the  equation 
(267)  may  therefore  be  readily  integrated,  and  will  give 


GRAVITATION.  265 

—M' 

replacing /Rc/r  by  this  value,  and  6— 2c  by  b',  the  equations 
(259)  and  (260)  will  become 

r^d^'+dr'  2M' 

V     ^^^^+7^ ^^^^^' 

,  adr 

^^r^{b'r2-a'+2Wr) ^      ^' 

506.  To  determine  the  value  of  the  constant  b',  or  its  equal 
6— 2c,  we  observe  that  the  equations  (258)  and  (268)  give,  by 
comparison,  * 

dx^-^-dy^  j'^dp^  ^dr^  __        2:vr 
dt^  dt^  ^  +    r   ' 

and  since  ^{dx""  ■\-dy'')  is  equal  to  ds,  the  element  of  the 

curve,  it  appears  that  the  quantity  — i- — - —  is  equal  to 

(ds\  ^ 
—  j   ,  or  equal  to  the  square  of  the  velocity  estimated  in 

the  direction  of  the  tangent  to  th^^  curve  ;  thus,  denoting  this 
velocity  by  v,  the  equation  (268)  will  become 

2M' 

,;2=:6'  +  f^ (270). 

If  V  represent  the  velocity  at  a  given  instant,  and  x  the  cor- 
responding value  of  the  radius  vector,  the  equation  (270) 
will  contain  but  a  single  unknown  quantity  b\  whose  value 
will  result 


507.  The  constant  a  may  also  be  determined  in  functions 
,  by  replacing 


of  the  initial  velocity  ;  for,  by  replacing  -—  in  the  formula 


dV 


) 


by  its  value  —  deduced  from  equation  (264),  we  shall  obtain 

v2=^+^ (271). 

dt=  ^r=  "^      ^ 

23 


266  DYNAMICS. 

The  quantity  dr  represents  the  infinitely  small  difference  ml 
{Fig.  189)  between  two  consecutive  radii  Am  and  Kn]  and 
by  regarding  the  triangle  mnl  as  rectilinear,  and  right-angled 
at  /j  we  shall  have 

7nl—mn .  cos  mnl, 

or, 

dr=ds .  cos  wmZ; 

(Is 
substituting  this  value  of  dr  in  (271),  and  changing  —  into  v, 

we  shall  find 

v^=v'^  cos'^  nml-i — . 

But  if  *  denote  the  value  of  the  angle  nml,  when  v  and  r 
are  transformed  into  V  and  a,  we  shall  have  the  relation 

Y''=Y^  .cos'  a-\-~: 

whence, 

«2=^272(1— cos^  «)=A2V2sin"'«; 
and  consequently, 

a=A.  V.sin«. 

508.  Having  determined  the  constants  which  enter  into 
equation  (269),  we  proceed  to  integrate  it,  for  the  purpose  of 
discovering  the  nature  of  the  trajectory  described  by  the  ma- 
terial point. 

To  facilitate  the  integration,  make  r=  -,  and  the  equation 

(269)  will  then  become 

■~~^[b'-{a''z='  -2M'z)]  ' 
or, 

adz 
d^= 


making 

az =p,  and  6'-t— — =Aa, 

a  a' 

the  preceding  equation  will  be  reduced  to 


GRAVITATION.  267 

,  _       —dp 

and  by  integrating,  we  find 

^+ constant = arc  |COS=^). 

Replacing  p  and  A  by  their  values,  suppressing  the  common 
factor  a,  and  denoting  by  4-  the  arbitrary  constant,  we  obtain 
/  a^z—W     \ 


whence. 


=cos(<?)  +  V')5 


and  by  restoring  the  value  of  z  in  terms  of  r,  we  finally 
obtain 

a=»— M'r=r^(a«6'+M'2)  .cos  (^+^^) (272). 

509.  The  arbitrary  constant  ^^  serves  merely  to  change  the 
direction  of  the  axis  with  which  the  radius  vector  forms  the 
variable  angle  :  if,  for  example,  the  angle  CAm  or  <(>  {Fig. 
190),  formed  by  the  radius  vector  with  the  primitive  axis  AC, 
be  supposed  successively  equal  to  1°,  2°,  3",  &c.  and  if  the 
variable  angle  be  reckoned  from  the  axis  AB,  which  forms 
with  the  axis  AC  an  angle  CAB=^^,  the  angle  included  be- 
tween the  radius  vector  km  and  the  axis  AB,  will  be  succes- 
sively equal  to 

1°+^^     2°+>/.,    3°+^/.,&c.; 
or,  in  general,  to 

510.  The  angle  <p-\-i^  will  disappear  from  equation  (272), 
when  the  polar  co-ordinates  are  transformed  into  rectangular 
co-ordinates,by  means  of  the  formulas 

r-  =xs  +y%    x==r  cos(^-l-'<^),    y—rsin(<p+4) (273)  ; 

for  the  first    two  of   these  formulas  reduce    the  equation 
(272)  to 

a^— MV(a^'+2/')=^\/(«'6'+M'2)  ; 
which  gives,  by  transposition, 

MV(^"+y')=a'-^\/(«'*'+M") (274): 

squaring  and  reducing,  we  find 


268  DYNAMICS. 

M'2y=  —b'a\v'  =a^  -2a^-a;^{a'b'  +  'M.'-) (275). 

This  equation  appertains  to  a  conic  section,  or  curve  of  the 
second  degree  :  it  will  be  the  equation  of  an  ellipse  or  hyper- 
bola, according  as  b',  upon  which  the  sign  of  the  second  term 
depends,  is  negative  or  positive  ;  for,  in  the  first  case,  the 
terms  containing  the  squares  of  the  co-ordinates  will  have 
similar  signs,  whilst  in  the  second,  tliey  will  be  affected  with 
contrary  signs  :  when  b'  becomes  equal  to  zero,  the  term  con- 
taining .?  -  will  disappear,  and  the  equation  will  then  apper- 
tain to  a  parabola. 

511.  If  we  resolve  equation  (275)  with  reference  toy,  there 
will  result 

y=±:^^y[a'-\-b'x'-2x^{a'b'  +  W)]; 

which  proves  that  every  rectangular  ordinate  is  equally 
divided  by  the  axis  of  x,  and  consequently  that  this  axis  must 
necessarily  be  the  greater  or  lesser  axis  of  the  curve  :  but  by 
introducing  into  equation  (274)  the  value  of  the  radius  vector 
given  by  the  first  of  equations  (273),  we  shall  obtain 

M'r = «2  _;r ^  (a=  6'  +  M'^  )  ; 
hence  it  appears  that  the  radius  vector  is  constantly  expressed 
in  rational  functions  of  the  absciss  x,  and  that  the  origin 
therefore  corresponds   to  the  focus.     Thus  the  co-ordinate 
axis  of  X  will  coincide  with  the  greater  axis  of  the  curve. 

512.  The  second  law  of  Kepler  is  thus  demonstrated  to  be 
a  consequence  of  the  hypothesis  of  Newton,  and  admits  of  a 
generalization  wholly  unknown  to  its  discoverer  ;  lie  was  in- 
duced, judging  by  analogy,  to  assign  elliptical  orbits  to  the 
planets,  whereas  it  appears  from  the  preceding  demonstration, 
that  they  might  have  described  either  hyperbolas  ox  parabolas. 
If  amongst  the  comets  hitherto  observed  we  have  found  no 
examples  of  a  hyperbolic  motion,  it  results  from  the  fact 
that  the  chance  of  a  body's  describing  a  curve  which  shall 
be  sensibly  hyperbolic  is  found  to  be  extremely  small.  "  I 
have  found,"  says  Laplace,  "  that  the  chances  are  at  least  six 
thousand  to  one  that  a  comet  which  comes  within  the  sphere 
of  the  sun's  action  will  describe  an  extremely  elongated 
ellipse,  or  a  hyperbola,  which,  by  the  magnitude  of  its  trans- 


GRAVITATION.  269 

verse  axis,  will  be  sensibly  confounded  with  a  parabola,  in 
that  portion  of  its  orbit  which  can  be  observed  ;  it  is  not  sur- 
prising, therefore,  that  the  hyperbolic  motion  has  not  yet 
been  observed." 

513.  If  in  equation  (275),  we  make  x=0,  and  y—\j*,  we 
shall  obtain  for  the  ordinate  passing  through  the  focus,  or 
the  semi-parameter, 

1   =^ 

514.  The  equation  (275)  admits  of  simplification,  by 
making 

^{a^h'-\-W')=n (276), 

and  transporting  the  origin  to  the  centre  of  the  curve  :  for  this 
purpose  we  make  x=x'-\-»^  and  dispose  of  the  arbitrary  quan- 
tity ct  by  the  condition  that  the  coefficient  of  the  first  power 
of  x'  shall  vanish.  Making  these  substitutions  in  equation 
(275),  and  dividing  by  a^ ,  we  find 

M'2                            ^         —  6'«=  i 
a^^  \x'     -f  2w<»  V  =0 (277). 

Putting  the  coefficient  of  x'  equal  to  zero,  we  have 

n 

6'' 
this  value  being  introduced  into  the  last  term  of  equation 
(277)  reduces  it  to 

y-"  ■ 

But  the  equation  (276)  gives 

substituting  this  value  for  the  last  term  of  equation  (277), 
and  suppressing  the  second  term,  which  by  hypothesis  is 
equal  to  zero,  we  shall  obtain 

or  by  clearing  the  denominators, 

h'^^y^—b'^a^x''^^a^W^Q (278). 


270  DYNAMICS. 

In  this  equation  the  origin  of  co-ordinates  is  at  the  centre 

of  the  curve  ;   hence,  if  we  make  ^=0,  and  deduce  the 

corresponding  value  of  x',  we  shall  have 

M' 
semi-axis  major =-^ (279)  ; 

and  by  making  a  similar  supposition  with  respect  to  x',  we 
find 

semi-axis  mmor=  w^  —  tt- 

This  value  becomes  imaginary  when  b'  is  positive,  agreeing 
with  the  -esult  in  Art.  510,  since  the  curve  described  is  then 
a  hyperbola  ;  but  the  value  is  real  when  b'  is  negative,  the 
curve  then  being  an  ellipse.  In  this  case^  if  we  replace  b'  by 
—  b',  we  shall  have 

semi-axis  minor= — — (280). 

515.  This  result  corresponds  with  that  which  would  have 
been  obtained  from  the  consideration  that  the  minor  axis  is  a 
mean  proportional  between  the  major  axis  and  the  parameter, 
the  values  of  which  have  been  already  obtained. 

516.  Having  determined   the  principal   elements   of  the 

curve  described,  it  will  now  be  easy  to  establish  the  third  of 

Kepler's  laws.     Let  tt  denote  the  number  3.1416  ;  then,  the 

area  of  an  ellipse  whose  semi-axes  are  represented  by  A  and 

B  will  be  expressed  by  ^AB  ;  and  if  A  and  B  be  replaced  by 

their  values  determined  in  equations  (279)  and  (280),  we 

shall  find 

waM' 
area  of  the  ellipse  described  by  the  planet=7; — jj  •  •  •  •  (281)  ; 

or, 
area  of  the  ellipse  described  by  the  planet =-^^(  ~  )  ^* 

But  it  has  already  been  shown  that  if  i  represent  the  time 
required  by  a  planet  to  describe  the  sector  JjAm  (Fig.  188), 
the  equation  (262)  will  give 

2  sector  LAm 
a 
When  t  becomes  the  time  of  an  entire  revolution,  which  we 


GRAVITATION.  271 

will  represent  by  T,  the  sector  hAm  will  become  the  area  of 
the  ellipse,  and  we  shall  then  have 


2^    /M'\l. 


M' 
and  since  —  represents  the  semi-axis  major,  we  shall  have, 

by  representing  its  value  by  D, 

or,  replacing  M'  by  its  value  (266),  we  obtain 

''-i^^) (^^^>- 

In  like  manner,  for  a  second  planet  m',  which  performs  its 
revolution  in  the  time  T',  in  an  ellipse  whose  semi-axis  major 
is  denoted  by  D',  we  shall  have,  since  the  mass  of  the  sun 
remains  invariable, 

3 

T/=-~?^l—^ (283)  : 

but  the  masses  of  the  planets  being  extremely  small  when 
compared  with  the  mass  of  the  sun,  we  may  neglect  the 
quantities  7Ji  and  m'  in  comparison  with  M  ;  and  the  equa- 
tions (282)  and  (283),  being  then  compared,  will  give  the 
proportion 

T  :  T'  :  :  D^  :  D'%     or  T^  :  T'^  :  :  D^  :  D'^  ; 

the  squares  of  the  times  of  revolution  will  therefore  be  pro- 
portional to  the  cubes  of  the  greater  axes  of  the  orbits  de- 
scribed, or  to  the  cubes  of  the  mean  distances  of  the  planets 
from  the  sun. 

517.  The  inverse  problem  may  also  be  resolved,  and  the 
law  of  gravitation  deduced,  from  the  elliptical  motions  of  the 
planets.  For  this  purpose,  we  must  adopt  the  hypothesis 
that  the  equation  (260)  refers  to  an  ellipse  :  but  the  polar 
equation  of  the  ellipse  being  of  the  form  Cr  cos  ç>=B^  —  Ar, 
its  differential  will  give 

*^^r^[(C2-A^)r»-B*  +2AB'r]' 


272  DYNAMICS. 

The  condition  of  identity  between  this  equation  and  equation 
(260)  requires  that  we  should  have 

— 7/Rrfr= AB2  =a  constant,    or  -fRdr ^^""^tant . 

differentiating,  and  suppressing  dr,  there  remains 
Tj  _  constant 

which  proves  that  the  force  varies  in  the  inverse  ratio  of  the 
square  of  the  distance. 


Of  the  Motions  of  Projectiles. 

518.  If  an  impulse  be  communicated  to  a  material  point  in 
a  direction  oblique  to  the  surface  of  the  earth,  the  point  being 
at  the  same  time  solicited  by  the  force  of  gravity,  it  will 
describe  a  trajectory,  the  nature  of  which  it  is  proposed  to 
investigate.  To  determine  the  circumstances  of  this  motion, 
we  will  denote  by  Aa-,  Ay,  and  Az  the  three  co-ordinate  axes , 
the  axis  Ajz  being  supposed  vertical.  The  force  of  gravity 
will  then  tend  to  diminish  the  co-ordinates  z  which  are 
reckoned  positive  upward,  and  if  its  intensity  be  supposed 
constant,  we  shall  have 

X=0,     Y=0,     Z=--. 
These  values  being  substituted  in  the  general  equations  (180) 
reduce  them  to 

d^x_f.     ^''y_n     d^z_^ 
'dF       '    JF      '     dF~~^'' 
the  first  two  of  these  equations  being  multiplied  by  dt,  and 
integrated,  give 

dx  dy     , 

dt  '  dt  ' 
the  constants  a  and  b  represent  the  velocities  of  the  material 
point  in  the  directions  of  the  axes  of  x  and  y  respectively. 
These  velocities  distinguish  the  motion  under  consideration 
from  that  which  takes  place  when  the  point  is  projected  ver- 
tically, their  values  in  the  latter  case  becoming  equal  to  zero. 
If  tiie  preceding  equations  be  multiplied  h^  dt,  and  again 
integrated,  we  shall  obtain 


PROJECTILES    IN    VACUO.  273 

x=at-\-a\     y=ht  +  h'\ 
and  eliminating  t  between  these  relations,  there  results 
hx  .  ab'  —  a'h 
a  a 

This  equation  appertains  to  a  right  line  EC  {Fig-  191),  situ- 
ated in  the  plane  of.r,  y,  and  the  trajectory  ELC  will  therefore 
be  contained  in  a  vertical  plane. 

519.  Since  the  trajectory  described  is  confined  to  a  vertical 
plane,  it  will  only  be  necessary  to  consider  the  two  co-ordi- 
nate axes  of  X  and  y,  the  former  being  supposed  horizontal 
and  the  latter  vertical  ;  we  therefore  employ  the  two  equations 

'dF       '     dr-~    '^' 
Multiplying  by  dt^  and  integrating,  we  find 

%=a,     'i^-St^r. (284). 

If  we  multiply  again  by  dt^  and  integrate,  we  shall  obtain 

.r=a^  +  a',     y—  —  \gt^  +ht-\-h' (285). 

To  determine  the  constants,  we  suppose  the  time  to  be 
reckoned  from  the  instant  at  which  the  material  point  leaves 
the  origin  of  co-ordinates  ;  whence, 

.r=0,    y=0,  and  ^=0; 
this  supposition  gives 

a'=0,     6'=0; 
and  the  equations  (285)  are  thus  reduced  to 

x—at^     y  —  —  \gt'^-\-ht. 
Eliminating  t  between  these  two  equations,  we  find 

y--a^-'^4' ('«")• 

The  equations  (284)  indicate  that  the  constants  a  and  h 
express  the  values  of  the  horizontal  and  vertical  compo- 
nents of  the  velocity  at  the  instant  from  which  the  time  is 
reckoned,  or  when  ^=0.  If,  therefore,  V  denote  the  initial 
velocity,  and  a  the  angle  formed  by  the  direction  of  the  initial 
impulse  with  the  axis  of  x^  the  componeYits  of  this  velocity 
will  be 

S 


274  DYNAMICS. 

V .  COS  «  parallel  to  the  eixis  of  x, 
V .  sin  «  parallel  to  the  axis  of  y  ; 

whence, 

a=V  cos  «,     6=V  sin  «. 

These  values  reduce  equation  (286)  to 

y=^  tang  .-^g^-^ (287). 

520.  This  equation  appertains  to  a  parabola,  having  its 
origin  at  the  point  A  {Ptg.  192),  the  vertex  being  situated  at 
a  point  C,  above  AB,  and  the  curve  extending  indefinitely 
below  AB  ;  for.  the  equation  (287)  being  of  the  form 

i/=mx — iix'^, 
by  making  y =0,  we  shall  obtain  for  the  abscisses  of  the  points 
at  which  the  curve  intersects  the  axis  of  x, 

x=0,   and  x= — . 
n 

7/1 

But  every  value  of  :r  less  than  —   will  give  a  positive  value 

for  7/,  whilst  every  value  greater  than  —  will  give  y  a  nega- 
tive value.  For,  if  we  multiply  by  nx  both  members  of  the 
inequality 

m 

x<—, 
n 

we  shall  obtain  nx^^mx,  the  condition  which  is  obviously 
necessary,  that  the  ordinate  y  may  be  positive.    In  like  man- 

771 

ner,  it  may  be  shown  that  when  .t>— ,  the  value  of  y  will 

n 

become  negative. 

521.  If /i  denote  the  height  from  which  a  body  must  fall  to 
acquire  the  initial  velocity  V,  we  shall  have  (Art.  401) 

V=^(2^A) (288): 

by  means  of  this  value,  the  equation  (287)  is  reduced  to 

x^ 

y=x  .  tang  » — — . — (289). 

^  ^         4/i  cos^'  «  ^       ' 

522.  The  distance  from  the  origin  A  to  the  point  B,  at 
which  the  curve  intersects  the  axis  of  a-,  is  called  the  range. 


PROJECTILES    IN    VACUO.  275 

To  determine  its  value,  we  make  y=0,  and  the  corresponding 
value  of  .r,  which  is  not  zero,  will  express  the  range.  Thus 
making  y=0,  in  (289),  we  have 

3;=0,   and  a:=4A.  tang  «.  cos^rtj 
the  second  value  of  x  gives,  by  reduction, 
x=-4Ji .  sin  a. .  cos  «  ; 
and  consequently, 

range =4/i .  sin  a .  cos  u, (290)  ; 

or,  replacing  2  sin  «  .  cos  »  by  its  equal  sin  2<«,  we  have 

range =2/i .  sin  2* (291). 

This  equation  may  be  employed  in  the  construction  of  tables 
which  shall  express  the  ranges  corresponding  to  different 
velocities,  and  different  angles  of  projection. 

523.  The  greatest  positive  ordinate  will  express  the  maxi- 
mum elevation  of  the  moveable  point  above  the  axis  of  x. 

To  determine  its  value,  we  make  -^=0;  or, 

ax 

-^  =tang  «■—— =0  ; 

dx         ^        2h  cos=^  a 

from  which  we  deduce 

x=2h .  cos2  «  .  tang  a, 
or, 

x=2h .  cos  « .  sin  «6  ; 
and  consequently,  the  absciss  of  the  highest  point  of  the  tra- 
jectory will  be  equal  to  one-half  the  range. 

Replacing  x  by  2/t .  cos  «  .  sin  «  in  equation  (289),  we  find 
for  the  maximum  elevation  of  the  moveable  point, 
y=/t .  sin^  ct. 

524.  The  projectik  may  be  impelled  in  two  different  direc- 
tions, so  as  to  produce  the  same  range.  For,  let  «'  represent 
an  angle  equal  to  the  complement  of  «  ;  the  equation  (290) 
will  give  the  value  of  the  range, 

4A .  sin  « .  cos  «=4/i .  sin  «  .  sin  «  . 
But  if  the  projectile  be  thrown  in  a  direction  forming  an 
angle  *'  with  the  axis  of  x,  the  range  will  be  expressed  by 
Ah  .  sin  «'.  cos  x=Ah  .  sin  «' .  sin  «. 
S2 


276  DYNAMICS. 

The  identity  of  these  expressions  for  the  ranges  corresponding 
to  the  angles  »  and  <*',  evidently  proves  that  the  ranges  will  be 
equal  when  the  two  angles  of  projection  are  complements  of 
each  other. 

525.  To  determine  the  angle  of  projection  which  corres- 
ponds to  the  greatest  range,  we  remark  that  the  range  is  in 
general  expressed  by  2h  sin  2«,  and  that  this  expression  will 
become  a  maximum  when  the  angle  2«  is  equal  to  90°  ; 
hence  it  follows  that  a  projectile  in  vacuo  will  have  the 
greatest  range  upon  a  horizontal  plane  when  the  angle  of 
projection  is  equal  to  45°. 

The  supposition  of  2<«=90°  gives  sin  2»=!  ;  consequently, 
the  expression  for  the  range  then  becomes  equal  to  2h  ;  or 
the  range  corresponding  to  the  angle  of  45°  is  equal  to  twice 
the  height  due  to  the  velocity  of  projection. 

Let  this  range  be  denoted  by  P  ;  we  shall  have 

h  =  \V (292). 

To  determine  the  value  of  the  coefficient  h,  the  projectile 
may  be  thrown  in  a  direction  forming  an  angle  of  45°  with 
the  horizontal  plane,  and  the  corresponding  range  may  then 
be  measured.  If  this  range  be  represented  by  P,  the  value 
of  h  will  immediately  result  from  equation  (292).  In  fire- 
arms, the  coefficient  h  serves  £is  a  measure  of  the  force  of  the 
powder,  since  the  extent  of  the  range  evidently  depends  on 
the  intensity  of  the  force  of  projection. 

526.  The  quantity  h  having  been  determined  by  taking 
the  mean  result  of  a  large  number  of  experiments,  we  substi- 
tute its  value  in  equation  (289),  which  will  thus  become 


'y=x  tanga- 


2P  cos-  « 

If  we  represent  by  P'  the  range  corresponding  to  an  angle  «', 
the  equation  (291)  will  give 

F=2^sin2«' (293); 

or, .replacing  h  by  its  value  iP  (292),  we  find 

P'=Psin2«'. 
This  relation  will  determine  the  range  P'  corresponding  to 
the  angle  «',  when  the  value  of  the  maximum  range  has 
been  previously  ascertained  ;  and,  in  general,  we  can  calcu- 


PROJECTILES    IN    VACUO.  277 

late  the  range  P'  which  corresponds  to  an  angle  «',  from,  a 
knowledge  of  the  range  P"  given  by  any  other  angle  «"  ;  for, 
since 

P'=P  sin  2«',     P"=P  sin  2«", 
we  obtain,  by  division, 

F  _sin  2cc'  ^ 

P"    sin  2<*"  ' 
if,  therefore,  the  range  P"  corresponding  to  the  angle  »"  be 
determined  by  measurement,  the  value  of  P'  corresponding 
to  X  will  result  immediately  from  the  preceding  equation. 

527.  The  value  of  h  (292),  being  substituted  in  equation 
(288),  will  give,  for  the  value  of  the  initial  velocity, 

V=v/(%)  =  v/(32ift.xP). 
If,  for  example,  the  range  corresponding  to  an  angle  of  45"^ 
were  equal  to  1000  feet,  we  should  find 

V=:^(1000  ft.x32ift.)  =  179.3ft.,  nearly. 

528.  If,  on  the  contrary,  the  initial  velocity  and  angle  of 
projection  were  given,  we  might  determine  the  range  :  for 
example,  let  the  initial  velocity  be  supposed  equal  to  200  feet 
per  second,  and  the  angle  of  projection  15"  ;  we  first  determine 
h  from  the  following  formula,  deduced  from  (288), 

V2     (200  ft.')* 
A=  — =  ^-— — ^  =  6^1  7  ft  • 
^     2g       64ift.       ^■^■^"•' 

and  the  range  P'  will  then  become,  (293), 

F=2x621.7ft.xsin30°=261.7ft. 

529.  The  problem  may  also  be  presented  under  the  follow- 
ing form  : — Having  given  the  initial  velocity  and  the  co-ordi- 
nates a:'=AB,  and  y'=BC,  of  a  point  C  {Fig.  193),  it  is  re- 
quired to  determine  the  angle  of  projection  such  that  the 
trajectory  may  pass  through  a  given  point  C.  The  equation 
V= v^(2g-A)  will  determine  the  value  of  h  ;  and  since  the  co- 
ordinates x'  and  y'  should  satisfy  the  equation  (287),  we  shall 
have  by  substituting  i/  and  y'  for  x  and  y, 

y'=x'  tang«----^^l— (294). 

^  ^        4A .  cos*  cc 

In  this  equation  the  quantity  «  is  alone  undetermined  :  mak- 
ing tang  ««=z,  we  have 


278  DYNAMICS. 

1  1 


COS  «=- 


sec*     ^(1+taiig^ct)     ^{i^z')' 
and  by  substituting  these  values  in  equation  (294)  we  find 

j/=x'.z-^{l+z=) (295). 

This  equation  being  resolved  with  reference  to  z,  will  give 
two  values  which  determine  the  two  angles  of  projection 
corresponding  to  the  directions  in  which  the  projectile  should 
be  thrown  in  order  that  it  may  strike  the  point  C  ;  we  select 
the  greater  of  these  two  angles  when  we  wish  to  crush  the 
object  upon  which  the  projectile  falls,  £is  the  vertical  velocity 
at  the  point  C  will  then  be  the  greatest. 

It  may  occur,  that  instead  of  the  line  CB,  we  have  given 
the  angle  CAB  subtended  by  the  object  CB.  Let  this  angle 
be  denoted  by  ^  ;  we  shall  have 

CB=:r'  tang^=y'; 
this  value  of  y',  being  introduced  into  equation  (295),  trans- 
forms it  into 

•  x' 
ta.ng<p=z——{l-{-z^)] 

from  which  we  deduce 

-'^±v/(^^-'-^f^-0- 

Of  the  Motions  of  Projectiles  in  a  Resisting  Medium. 

530.  The  theory  of  projectiles  in  vacuo,  which  has  been 
examined  in  the  preceding  paragraphs,  afibrds  results  which 
diifer  greatly  from  those  obtained  by  direct  experiments  per- 
formed in  the  atmosphere  :  these  discrepances  are  very  con- 
siderable when  the  velocity  of  projection  is  great,  and  are  to 
be  attributed  to  the  resistance  opposed  by  the  atmosphere  to 
the  motion  of  a  body.  If  this  resistance,  represented  by  R,  be 
supposed,  as  in  Art.  412,  to  vary  in  the  duplicate  ratio  of  the 
velocity,  we  shall  have 

The  resistance  R  at  each  point  of  the  trajectory  will  be 
exerted  in  the  direction  of  the  element  of  the  curve,  but  in  an 


PROJECTILES    IN   A    RESISTING    MEDIUM.  279 

opposite  direction  to  that  of  the  motion  ;  and  the  force  R 
will  form  with  the  axes  of  co-ordinates  the  same  angles  as  the 
element  ds.  Thus,  denoting  by  «,  |3,  and  y  the  angles  included 
between  the  tangent  to  the  curve  at  any  point  and  the  co- 
ordinate axes,  the  components  of  R  will  be  expressed  by 

R  cos  »,    R  cos  /3,     R  cos  y. 
To  obtain  expressions  for  these  cosines,  let  mm'  {Pig-  194) 
represent  an  element  ds  of  the  curve  :  the  projection  of  this 
element  on  the  axis  of  z  will  be  equal  to  m'n.     But  the  tri- 
angle 7}i'mn  gives  the  proportion 

1  :  cos  mm'n  :  :  mm/  :  m'n  ; 
or, 

1  :  cos  y  ::  ds  :  dz] 

hence, 

dz 

cos  y^^-j-  '} 
as 

and  the  component  of  R  in  the  direction  of  the  axis  of  z,  will 
therefore  be  expressed  by 

R— 

ds' 

We  attribute  the  negative  sign  to  this  component,  because  the 
tendency  of  the  force  R,  while  the  projectile  is  moving  from 
9)1  to  m',  will  be  to  diminish  the  co-ordinate  z.  Fora  similar 
reason  the  other  components  of  the  resistance  R  should  be 
affected  with  the  negative  sign. 

531.  An  analogous  course  of  reasoning  will  give 

— R— -  for  the  component  of  R  in  the  direction  of  the  axis  ofx, 
ds 

— R—  for  the  component  in  the  direction  of  y. 
ds 

Thus,  the  equations  expressing  the  circumstances  of  the 

motion  will  be 

d^x    -odx 

'di^~~~    ds' 

^  =  _R^ 

dt^  ds* 

d^z_     j^dz 

'dF — ^d^~^' 


280 


DYNAMICS. 


From  the  first  two  we  obtain,  by  division, 
d^y  _dy  ^ 

or, 

S='i^ <^^«)^ 

and  by  integration, 

log  dy~\og  dx-\-\og  a=log  adx. 
Passing  from  logarithms  to  numbers,  we  find 

dy^adx  ; 
and  by  a  second  integration, 

y=ax  +  b  ; 
hence  we  conclude,  that  the  projection  of  the  trajectory  on 
the  plane  of  a:-,  y  is  a  right  line,  and  therefore  that  the  trajec- 
tory is  contained  in  a  vertical  plane. 

532.  If  we  resume  the  consideration  of  the  problem  with 
this  restriction,  that  the  curve  shall  be  confined  to  a  vertical 
plane,  it  will  only  be  necessary  to  employ  the  two  equations 

d^x_     Tydx      d^y  _     -^dy 
dF  ds'     ~dt^  ds~~^' 

It  has  already  been  remarked,  that  the  vertical  component  of 

the  resistance  R-r^  should  be  affected  with  the  negative  sign, 

since  this  resistance  tends  to  diminish  the  co-ordinate  ;  but 
this  tendency  will  only  exist  whilst  the  projectile  is  describing 
the  ascending  branch  of  the  trajectory.  If,  on  the  contrary, 
the  projectile  be  supposed  at  a  point  M"  in  the  descending 
branch  {Pig-  194),  the  resistance,  being  exerted  in  the  direc- 
tion from  M"  to  M',  would  tend  to  increase  the  co-ordinate  y. 

It  might,  therefore,  appear  that  the  component  R— should 

change  its  sign  ;  but  since  dy  becomes  negative  in  the  second 
branch  of  the  curve,  the  vertical  component  will  still  be  ex- 
pressed by  —  R  J^, 

If  the  quantity  R  in  the  preceding  equations  be  replaced 
by  its  value  mv"^ ,  they  will  become 


PROJECTILES    IN    A    RESISTING    MEDIUM.  281 

d^x  „dx     d^y  Ay 

df^  ds'     dt""  ds     ^ 

The  quantity  v^  may  be  eliminated  by  means  of  the  equation 

ds^ 


dp' 


and  we  shall  have 


IF-    "^W^Ts ^'^^^^' 

^=_^^xf^-^ (298). 

dp  dt""     ds     ^  ^      ' 

533.  The  first  of  these  equations  being  multiplied  by  dU 

gives 

d^x  J      dx    ds 

—-—=—mds  .—-.-— 

dt  ds     dt 


or. 


d^x  J  dx 

dt  dt  ' 


from  this  equation,  we  deduce 
d^x 
dt 


=  —mds\ 


dx 
W 

and  by  integration, 

log  —  =  — ms-\-C. 

534.  Let  A  represent  the  number  whose  logarithm  is  equal 

to  C,  and  e  the  base  of  tl^e  Naperian  system  ;  we  shall  have 

C=log  A,    log  e=l  ; 

the  preceding  equation  may  therefore  be  transformed  into 

dor 
log  -z-  =  — ms  log  e+log  A, 

or^ 

log^  r^log  e-™+log  A=log  Ae-"-î 
dt 

passing  from  logarithms  to  numbers,  we  have 

$=Ae-^ (299). 

dt 


282  DYNAMICS. 

535.  To  determine  the  constant  A,  let  V  represent  the 
initial  velocity,  and  «  the  angle  formed  by  the  direction  of  the 
initial  impulse  with  the  axis  of  w.  The  component  of  V  in 
the  direction  of  this  axis  will  be  expressed  by  V  cos  «.     But 

when  5=0,  -^  will  express  the  comp  'Uent  of  the  initial  velocity 

along  the  axis  of  a-  :  hence  the  preceding  equation  will,  on  this 
supposition,  be  reduced  to 

V  cos«=Ae°=A. 
This  value  substituted  in  (299)  converts  it  into 

^= V  cos  u  .  e-"" (300). 

at 

536,  Since  this  equation  contains  three  variables,  we  must 
obtain  a  second  relation  between  them,  in  order  to  render  the 
integration  possible.  For  this  purpose,  the  equations  (297) 
and  (298)  may  be  written  under  the  form 

dx_  ~  'dP        dy_~\dP^) 

ds  ds^^     lis  dT^       ' 

dV"  dV 

the  quantity  ds  may  be  eliminated  immediately  by  division  ; 

and  we  thus  obtain 

d^y 

dy_dt^      ^ 

dx       d^x 

w 

From  this  equation  we  deduce 

_dy  d^x _d^y , 
^~di''dF~dF^ 
or,  by  reduction, 

Jyd^.x-dxd^y 3Q^^ 

dx 
The  second  member  being  divided  by  —dx  becomes  the  ex- 
act differential  of  -^  ;  and  the  equation  (301)  may  there- 
at: 

fore  be  written 


PROJECTILES    IN    A    RESISTING    MEDIUM.  283 

(lu 

If,  for  greater  simplicity,  we  make  j~  =zp,  there  will  result 

gdP^-dx.dp (302); 

and  eliminating  dt,  by  means  of  equation  (300),  we  find 

^=-V2  cos^^  *  .  e-^™  .  ^ (303). 

537.  This  equation  still  contains  three  variables  ;  but  one 
of  them  may  be  readily  eliminated  by  means  of  the  relation 
ds=^{dx^  -\-dy-)  ;  in  which,  replacing  dy  by  its  value  pdx, 
we  obtain 

ds^dx^il+j^"") (304); 

and  consequently,  by  eliminating  dx  between  this  equation 
and  (303),  there  will  result 

dp^(^+p')=^S^ (305). 

Litegrating,  we  have 

i/V(l+7^^)+|log[i^  +  x/(l+P^)]=C-^^J^^...(306); 

and  by  making  C=iB,  and  suppressing  the  common  divisor 
2,  we  obtain 

2V(l+2^^)+log  b+^/a+P=')]=B-^J^^^^ (307). 

To  determine  the  value  of  the  constant  B,  we  observe  that 

—  expresses  the  trigonometrical  tangent  of  the  angle  formed 
dx 

by  the  element  of  the  curve  with  the  axis  of  x.  At  the  point 
A,  the  origin  of  the  motion,  this  angle  is  denoted  by  * ,  the 
quantity  t  being  at  the  same  time  equal  to  zero  ;  we  shall 
therefore  have 

:r=0,     y=0,     s=0,    jo=tang«. 
These  values  of  s  and  jt  being  substituted  in  the  preceding 
equation  give 

B=tang  »v^(14-tang2*) 

+log[tang.+  v/(l+tanr*)]+^^^VWi:  ' 
the  value  of  the  constant  B  in  equation  (307)  may  therefore 
be  regarded  as  known. 


284  DYNAMICS. 

538.  If  we  eliminate  e'^™  between  the  equations  (303)  and 
(307),  we  shall  obtain 

dx= ^ = (308). 

The  two  members  of  this  equation  being  multiplied  by  the 
corresponding  members  of  the  equation 

dy 

there  will  result 

dy= = A ^.^^ (309). 

m[py/l-]-p''  +log  (p  +  \/l+p^)— B] 

539.  To  determine  the  time  t,  we  substitute  in  the  equation 

dt^  =  -±L^, 
g 
the  value  of  dx,  given  by  equation  (308),  and  we  thus  obtain 

dt''  = — ^:^ (310)  ; 

m^[jo^/l+p2+log(Jo  +  ^/l4-232  )— B] 

or,  by  changing  the  signs  of  the  numerator  and  denomi- 
nator, 


dt^z= 


dp'- 


mg[—pVl+p^—\og{p  +  V\+p'')-\-B] 
In  extracting  the  square  root  of  the  two  members  of  this 
equation,  the  second  might  be  affected  with  the  double  sign, 
but  in  the  present  instance  we  shall  attribute  to  it  the  nega- 
tive sign.  For,  since  every  equation  between  two  variables 
t  and  p  may  be  regarded  as  that  of  a  curve,  of  which  t  is  the 
absciss,  and  j»  the  ordinate,  Up  increases  whilst  t  diminishes, 
the  elements  dt  and  dp  will  necessarily  be  affected  with  con- 
trary signs.  But,  in  the  present  case,  it  is  obvious  that  whilst 
t  augments,  the  quantity  p,  which  expresses  the  trigono- 
metrical tangent  of  the  angle  formed  by  the  element  of  the 
curve  with  the  axis  of  x^  constantly  diminishes  in  the  ascend- 
ing branch  of  the  trajectory,  which  is  the  one  at  present  under 
consideration;  hence,  we  shall  have 

dt= ^^  __ (311). 

y/mg[-2}y/l-\-p^—\og{p+\^l-\-p'')-{-B] 


PROJECTILES    IN    A    RESISTING    MEDIUM.  2S5 

540.  The  expression  for  the  velocity  can  now  be  obtained 
in  functions  of  jo  ;  for,  the  velocity  resulting  from  the  equation 
ds     ^(dx^  -\-dy-)     dx     .^  ,      . 

we  obtain,  after  replacing  dx  and  dt  by  their  respective 
values, 


V: 


|xwi+i>^) 


\/-/>v/(l+P=')-logb  +  v^(l+P')]+B 
541.  We  can  also  express  the  arc  s  in  functions  oip  ;  for 
the  equation  (307)  gives 

Taking  the  logarithms,  and  reducing,  we  obtain 
'mV-cos^  <* 


1      {' 


[B— _p%/l+p2— log  (p  +  v/l+jo*)] 


2711 

542.  To  obtain  the  equation  of  the  trajectory,  it  would  be 
necessary  to  integrate  equations  (308)  and  (309)  :  these  inte- 
grations cannot  be  effected  except  by  the  aid  of  series.  Never- 
theless, by  employing  equations  (308)  and  (309),  the  curve 
may  be  constructed  approximatively  by  points. 

For  this  purpose,  we  will  write  those  equations  under  the 
form, 

dx=<pj)  .dp (312), 

dy=^p .dp (313)  ; 

in  which  <pp  and  ■<l'p  represent  certain  known  functions  of». 
The  first  of  these  equations  gives 
dx 

dx 
the  quantity  —  represents  the  tangent  of  the  angle  included 

between  the  axis  of  abscisses  and  the  element  of  a  curve 
whose  co-ordinates  are  denoted  by  p  and  x  respectively.  We 
will  first  construct  this  curve,  which  will  serve  to  determine 
points  in  the  trajectory.  It  is  distinguished  by  the  name  of 
the  auxiliary  curve. 


286  DYNAMICS. 

Having  drawn  two  rectangular  axes  A^  and  Ax  {Pig-  195), 
lay  off  from  A  to  B  a  distance  AB=tang«  ;  the  point  B  will 
appertain  to  the  auxiliary  curve,  since  the  ordinate  x=0 
corresponds  to  the  absciss  p=ta.nga.' 

If  the  line  AB  be  divided  into  equal  parts  BB',  B'B",  âcc.^ 
each  of  these  parts  being  represented  by  dp,  it  will  be  easy  to 
construct  approximatively  the  points  M,  M',  M",  <fec.  of  the 
auxiliary  curve,  corresponding  to  the  points  B,  B',  B",  <fcc. 
For,  if  we  suppose  the  points  B,  B',  B",  &.c.  to  be  exceedingly 
near  to  each  other,  we  may  regard  the  arcs  M'B,  M"M', 
M"'M",  &c.  of  the  curve  as  coinciding  with  the  tangents 
drawn  to  the  points  M',  M",  M'",  &c.  The  ordinates  M'B', 
M"B",  M"'B"',  &c.  may  then  be  calculated  ;  for,  the  trigono- 
metrical tangent  of  the  angle  formed  by  the  element  of  the 

dv 
curve  with  the  axis  of  p,  being  represented  in  general  by  J~ , 

its  value  will  always  be  given  by  means  of  equation  (312)^ 
whenever  we  assume  a  value  for  p.  Thus,  if  we  wish  to 
determine  the  trigonometrical  tangent  of  the  angle  WBp  in- 
cluded between  the  tangent  at  M',  and  the  axis  of  ab- 
scisses, since  the  absciss  of  the  point  M'  is  AB'=:AB — BB'= 
tang  cc — dp,  it  will  be  necessary  to  change  p  into  tang  d—dp, 

dx 
in  the  value  <pp=--,  given  by  equation  (312)  :   we  thus 

obtain 

tang  WBp=Ç){iQXig  a.— dp)  ; 
whence, 

tang  M'BB'=— ^(tang  ^-dp). 

The  ordinate  M'B'  being  expressed  by  BB'  X  tang  M'BB',  we 
shall  have 

M'B'=BB'x tang  M'BB'; 
or, 

WB!=dp  X  — ^(tang  »—dp). 

Thus,  the  point  M',  of  the  auxihary  curve  BC,  may  be  con- 
structed by  means  of  the  co-ordinates 

AB'=tang  a. — dp, 
and 

BW=dp  X  —  ^(tang  a— dp). 


PROJECTILES    IN   A    RESISTING    MEDIUM.  287 

To  determine  a  third  point  M",  we  make  AB"=tang  a.—2dp  ; 
and  by  the  same  course  of  reasoning  prove  that  the  trigo- 
nometrical tangent  of  the  angle  M"M'0  is  expressed  by 
— ^(tang  *— 2rfp),  and  consequently, 

WO=dp  X  -^(tang  ^—2dp) 
substituting  this  value  and  that  of  M'B'  in  the  equation 

M"B"=M'B'  +  M"0, 
given  by  an  inspection  of  the  figure,  we  find 

]\|"B"=  —dp  .  ^(tang  it— dp)  —dp  .  <Z)(tang  a—2dp). 
To  calculate  the  ordinate  M"'B"'  which  corresponds  to  the 
absciss  AB"'=tang  *— Srfja,  it  will  only  be  necessary  to  add  to 
the  value  of  M"B"  that  of  the  portion  M"'0',  which,  by  an 
investigation  similar  to  the  preceding,  may  be  proved  equal 
to  — dp  .  (f(tangct—  Mp)  :  thus  we  have 

B"'M"'=  —dp .  ^(tang  <t—dp)—dp  .  ^(tang  a.—2dp) 

— dp  .<p{t3Lng  cc—3dp). 

In  this  manner  we  may  determine  a  series  of  points  which 
will  appertain  to  the  auxiliary  curve,  the  co-ordinates  of 
which  are  .v  .  nd  p.  Connecting  these  points  by  right  lines, 
we  form  a  polygon  BM'M"M"',  &c.,  which  will  coincide  more 
nearly  with  the  curve,  in  proportion  as  dp  has  a  smaller 
value  assigned  to  it. 

By  performing  similar  operations  with  reference  to  the 
equation 

di/=4^p  .  dp, 
we  may  construct  a  second  auxiliary  curve  BD,  the  co-ordi- 
nates of  which  will  represent  the  quantities  p  and  y.  The 
co-ordinates  mb  and  /'6,  which  in  these  two  curves  correspond  to 
the  same  value  of  p,  will  represent  the  two  co-ordinates  of  a 
point  in  the  trajectory  ;  so  that  by  taking  the  co-ordinates 
B'M',  B"M",  B"'M"',  &c.  of  the  first  curve  as  the  abscisses 
of  the  trajectory,  its  ordinales  will  be  represented  by  the  lines 
B'L',  B"L",  B"'L"',  &c. 


288  DYNAMICS. 


Of  the  different  Methods  of  measuring  the  Effects  of  Forces. 

543.  It  has  been  remarked  (Art.  388),  that  two  forces  F 
and  F'  applied  to  the  same  body  are  proportional  to  the  veloci- 
ties which  they  can  impress  upon  that  body.  Let  it  now 
be  supposed  that  these  forces  are  applied  to  different  maisses. 

If  two  equal  forces  acting  in  opposite  directions  be  applied 
to  equal  and  spherical  masses  M  and  M',  they  will  commu- 
nicate to  these  masses  the  equal  velocities  V  and  V  ;  and  if 
these  masses  be  supposed  to  impinge  directly  upon  each 
other,  they  will  mutually  destroy  each  other's  motion,  and  an 
equilibrium  will  ensue,  since  the  circumstances  of  motion  in 
each  are  precisely  similar.  But  if  the  mass  M  be  supposed 
equal  to  nW,  and  V  greater  than  V,  we  may  regard  M  as 

composed  of  n  masses  m',  m",  m'", m'"',  each  equal  to 

the  mass  M'.  In  consequence  of  the  mutual  connexion  of 
the  different  parts  of  the  system,  each  of  the  nicisses  ni',  m", 
wl'\  (fee.  must  move  with  the  same  velocity  V,  so  that  if  the 
body  M  be  supposed  to  piss  over  the  space  of  three  feet  in  one 
second  of  time,  each  of  the  masses  m',  ml\  m'",  (fee.  will  like- 
wise pass  over  a  distance  of  three  feet  in  one  second  ;  or,  if 
V  represent  the  velocity  of  the  mass  M,  V  will  likewise  ex- 
press the  velocity  of  each  of  the  masses  m',  w",  m'",  (fee.  But 
if  the  mass  m',  moving  with  the  velocity  V,  should  impinge 
against  the  equal  mass  M',  which  moves  with  the  velocity  V, 
it  would  destroy  a  portion  of  the  velocity  of  the  second  body 
equal  to  V  ;  and  if,  at  the  same  instant,  the  mass  m",  acting  by 
its  connexion  with  the  other  masses,  should  impinge  against 
the  body  M',  it  would  likewise  destroy  a  portion  of  the  velo- 
city V,  equal  to  V  :  and  the  same  may  be  said  of  the  other 
masses  m!'\  rrû%  (fee.     Thus,  the  joint  effect  of  the  several 

masses  m',  m",  m"', m*"\  would  be  to  destroy  in  the 

mass  M'  a  velocity  represented  by  iiN.  If  we  suppose  the 
velocity  V  to  be  entirely  destroyed,  an  equilibrium  will  ensue, 
and  it  will  be  necessary  that  V'=?iV. 

By  eliminating  n  between  this  equation  and  the  relation 
M=wM',  we  obtain  the  proportion 

M:M'::  V:  V; 


MEASURE    OF    FORCES.  289 

from  which  we  concUide  that  an  equilibrium  will  ensue  when 
tv)o  bodies  are  caused  to  impinge  directly  against  each  otlier^ 
with  velocities  inversely  proportional  to  their  masses. 

544.  It  may  be  readily  demonstrated  that  the  same  propo- 
sition is  equally  true  when  the  mass  M  does  not  contain  the 
mass  M'  an  exact  number  of  times.  For,  if  the  mass  M  be 
supposed  to  contain  'n  masses,  each  of  which  is  equal  to  m^ 
and  the  mass  M'  to  contain  a  number  of  these  equal  masses, 
denoted  by  n'  ;  each  mass  m  contained  in  M,  will  destroy  a 
portion  V  of  the  velocity  V  of  a  mass  m  contained  in  M'  ;  or, 
since  M'  is  supposed  to  contain  n'  masses,  each  of  whicli  is 
equal  to  m,  the  mass  m,  moving  with  the  velocity  V,  will  de- 

V 
stroy  in  M'=m'/w  a  velocity  expressed  by  —  :    and  since   the 

other  equal  masses  contained  in  the  body  M  will  produce 
similar  effects,  the  entire  velocity  destroyed  in  M'  by  M  will 

V 

be  equal  to  —  repeated  as  many  times  as  the  mass  m  is  con- 
it' 

V 

tained  in  M,  or  it  will  be  equal  to  —  Xw:  if  we  suppose  the 

velocity  V  to  be  entirely  destroyed,  we  must  have 

V— V*il  • 

or, 

V  :  V  ::/«':  w  :  :  mn'  :  tnn  ; 
and  replacing  mn,  mn',  by  their  values  M,  M',  we  obtain  the 
proportion 

V  :  V  :  :  M'  :  M  : 

whence  the  truth  of  the  proposition  is  manifest. 

545.  Since  the  masses  of  the  bodies  are  in  the  inverse  ratio 
of  their  velocities  when  an  equilibrium  is  produced,  it  follows, 
that  if  the  bodies  have  equal  volumes,  and  unequal  densities, 
their  velocities  will  be  in  the  inverse  ratio  of  their  densities. 

546.  Let  F  represent  a  force  which  impresses  a  velocity  V 
upon  a  meiss  M  :  if  the  same  force  be  supposed  to  act  upon  a 
mass  M  times  less,  and  which  will  consequently  be  repre- 
sented bv  — =1,  this  force  will  communicate  to  the  mass 

'  M 

T  25 


290  DYNAMICS. 

unity,  a  velocity  M  times  greater  than  that  communicated  to 

the  mass  M  :  this  velocity  will  therefore  be  expressed  by  MV. 

For  a  similar  reason, the  force  F',  which  communicates  to  the 

mass  M'  a  velocity  V,  would  communicate  to  the  mass 

M' 

—  =  1,  a  velocity  represented  by  M'V. 

The  velocities  represented  by  MV  and  M'V  being  com- 
municated by  the  forces  F  and  F'  to  the  mass  unity,  it  follows, 
from  the  principles  enunciated  in  Art.  388,  that  we  shall  have 
the  proportion 

F  :  F'  :  :  MV  :  M'V. 

The  expressions  MV  and  M'V  are  called  the  quantities  of 
^notion  communicated  by  the  forces  F  and  F'  ;  and  it  should 
be  recollected  that  the  characters  M,  V,  F,  M',  V,  and  F' 
represent  abstract  numbers,  which  merely  express  the  number 
of  times  which  the  quantity  under  consideration  contains  the 
unit  of  its  own  species. 

547.  The  unit  of  force  being  arbitrary,  we  may  represent 
it  by  the  quantity  of  motion  which  it  produces.  Thus,  by 
supposing  F'to  represent  this  unit,  we  can  replace  F'by  M'V 
in  the  preceding  proportion  ;  and  we  thence  infer  that 

F=MV. 

548.  When  the  force  <p  acts  incessantly,  it  has  been  shown, 
Art.  388,  that  this  force  will  be  represented  by  the  velocity 
which  it  would  communicate  in  a  unit  of  time,  if  the  value 
of  the  force  should  become  constant  ;  hence  we  obtain,  by 
substituting  for  V  its  value  «p, 

F=M^. 
If  the  mass  M  be  supposed  equal  to  unity,  we  shall  have 

F=^; 
consequently,  (p  represents  the  force  exerted  upon  the  unit  of 
mass  ;  the  quantity  4>  is  usually  called  the  acceleratiJig  force, 
and  F  is  called  the  waving  force.  When  F  is  given,  the 
value  of  ^  can.be  determined  by  simply  dividing  by  M,  the 
mass  moved. 

549.  It  has  been  shown.  Art.  163,  that  if  g  represent  the 
force  of  gravity,  P  the  weight  of  the  body,  and  M  its  mass, 
we  shall  have 


MEASURE   OF    FORCES.  291 

eliminating  M  between  this  equation  and  the  preceding,  there 
results 

g 
and  if  the  incessant  force  <p  be  that  of  gravity,  we  have  <p  =g  ; 
hence, 

F=P; 

and  in  this  case  the  moving  force  is  measured  by  the  weight 
of  the  body  upon  which  the  force  is  exerted. 

550.  The  writers  upon  Mechanics  were  long  divided  in 
opinion  as  to  the  proper  measure  of  forces.  TJiis  disagree- 
ment, like  many  others,  arose  entirely  from  a  misapprehen- 
sion of  the  signification  of  words. 

The  nature  of  forces  being  known  to  us  only  by  the  effects 
which  they  produce,  we  may  with  propriety  measure  these 
eifects  in  different  ways,  according  to  the  object  which  it  is 
desired  to  accomplish.  If,  for  example,  it  be  proposed  to 
determine  the  load  which  a  man  can  support  for  an  instant 
of  lime,  it  is  evident  that  the  force  exerted  by  the  man  will 
be  proportional  to  the  weight  which  he  can  sustain,  and  may 
therefore  be  measured  by  this  weight  :  but  if  we  wish  to 
measure  the  force  of  this  man  by  the  work  which  he  can 
perform  in  a  given  time,  we  must  adopt  a  measure  for  the 
force  entirely  different  from  the  preceding  :  for,  it  might 
happen  that  a  man  absolutely  weaker,  but  endued  with  a 
greater  capacity  of  sustaining  a  continued  effort,  would  give 
by  his  labour  a  result  greater  than  that  given  by  the  first 
man,  and  might  therefore  be  considered  as  actually  possessed 
of  greater  force. 

In  this  second  method  of  considering  the  effects  of  forces, 
we  regard  them  as  proportional  to  the  weight  raised,  and  the 
height  to  which  it  is  elevated  in  a  given  time  ;  it  being  always 
understood  that  the  effort  necessary  to  overcome  the  weight 
is  not  supposed  to  vary  with  the  elevation. 

If,  for  example,  two  men  raise  the  same  weight,  in  the  same 
time,  to  the  heights  of  600  and  200  yards  respectively,  we 
would,  according  to  this  method  of  estimating  the  effects  of 

T2 


292 


DYNAMICS. 


forces,  reo^ard  the  first  as  possessed  of  three  times  the  force  of 
the  second. 

Again,  if,  in  the  working  day,  one  man  can  raise  a  weight 
of  50  lbs.  through  a  height  of  200  yards,  and  a  second  a  weight 
of  251bs.  through  a  height  of  400  yards,  we  should  regard  the 
two  men,  according  to  the  present  hypothesis,  as  possessed  of 
equal  strength,  although  the  absolute  strengths  of  the  two 
might  be  very  different  ;  the  strengths  of  the  two  individuals 
are  here  considered  only  with  reference  to  the  work  done. 

This  method  of  estimating  forces  was  adopted  by  Descartes. 
The  difference  in  the  opinions  entertained  by  him  and  other 
geometers  rested  entirely  on  the  definition  of  the  word /orce. 
He  contended  that  a  force  should  be  measured  by  the  product 
of  the  mass  into  the  square  of  the  velocity.  This  conse- 
quence may  be  deduced  from  the  definition  of  the  effect  of  a 
force,  adopted  by  Descartes,  in  the  following  manner. 

Let  P  represent  a  weight,  and  h  the  height  to  which  it  can 
be  raised  in  a  given  time  :  the  force  employed  to  raise  it,  ac- 
cording to  the  definition  of  Descartes,  will  be  measured  by 
the  product 

PX/i. 
We  can  replace  P  in  this  expression  by  its  value  M^  (Art. 
163),  and  we  shall  have 

?h=mgh  ; 

or,  multiplying  by  2, 

2PA=Mx2^A; 
and  since  the  velocity  v  due  to  the  height  h  is  expressed  by 
i/{2gh)  (Art.  401),  the  preceding  expression  becomes 

2PA=My^ 
Having  given  a  definition  of  the  word  force  difierent,  from 
that  adopted  by  Descartes,  we  shall  not  say  that  the  force  is 
measured  by  the  product  Mv^*,  but  that  it  is  measured  by  the 
quantity  of  motion  Mv  whieli  it  is  capable  of  producing,  as 
has  been  explained  in  Art.  547  ;  and  to  avoid  confusion,  we 
shall,  according  to  ordinary  usage,  apply  the  term  living  force 
to  the  product  M.v^,  of  the  mass  by  the  square  of  the  velocity. 
551.  The  consideration  of  living  forces  is  of  great  utility 
in  estimating  the  effects   produced  by  a  machine.     Thus,  if 


COLLISION  OP    UNELASTIC    BODIES.  293 

it  were  required  to  calculate  the  eifect  of  a  given  fall  of  water, 
the  force  necessary  to  move  a  carriage  on  a  given  piece  of 
ground,  or  the  eflfort  requisite  to  raise  a  given  mass  of  coals 
from  the  bottom  of  a  mnie,  we  might  in  each  case  compare 
the  effect  of  the  moving  force  to  the  product  of  a  certain 
weight  by  a  given  height,  or  to  an  expression  of  the  form  P/i, 
the  double  of  which,  as  has  been  before  shown,  is  equivalent 
to  the  product  Mv'^. 

Of  the  Direct  Impact  of  Bodies. 

552.  Bodies  are  usually  distinguished  as  elastic  or  unelastic. 
An  elastic  body  is  that  which,  when  compressed  by  the  appli- 
cation of  an  impulse,  will  resume  its  original  figure  with  a 
force  equal  to  that  of  compression,  in  virtue  of  a  quality  pos- 
sessed by  the  body.  An  unelastic  body,  on  the  contrary,  is 
one  whose  figure  either  undergoes  no  change  by  the  action 
of  a  force  applied  to  it,  or  which,  if  compressed,  has  no  tendency 
to  restore  itself  to  its  original  form. 

All  natural  bodies  are  found  to  partake  more  or  less  of  these 
two  qualities  ;  there  being  none  which  are  perfectly  elastic, 
or  perfectly  unelastic. 

Of  the  Direct  Impact  of  Unelastic  Bodies.. 

553.  Let  M  and  M'  {Pig.  196)  represent  two  spherical  un- 
elastic bodies,  which  move  in  the  direction  from  A  to  0.  If 
the  velocity  of  M  be  supposed  to  exceed  that  of  M',  the  former 
will  overtake  the  latter,  and  will  communicate  to  it  a  portion 
of  its  motion,  until  the  velocities  of  the  two  bodies  become 
equal.  Let  F  and  F'  represent  the  forces  which  communicate 
to  the  bodies  M  and  M'  their  respective  velocities  V  and  V  ; 
since  these  forces  can  be  represented  by  the  quantities  of  mo- 
tion which  they  produce  (Art.  547),  we  shall  have 

F=MV,     F'=M'V'; 

and  by  compounding  these  two  forces,  their  resultant  will  be 

expressed  by 

F  +  F=MV-fMV. 


294  DYNAMICS. 

To  obtain  a  second  expression  for  F+F',  let  v  represent  the 
common  velocity  of  the  two  bodies  after  impact  :  we  may 
regard  the  mass  M  +  M'  a^  a  single  body,  to  which  the  velocity 
V  has  been  imparted  by  the  exertion  of  a  force  F+F'.  We 
shall  then  have 

F+F=(M+M')v. 
By  equating  these  two  values  of  F  +  F',  we  obtain 

(M  +  M')î;=MV  +  M'V'; 
whence,  we  deduce 

_MY  +  M'V 
'"~    M  +  M'   ■ 
554.  If  the  bodies  move  in  opposite  directions,  we  regard 
one  of  the  velocities  V  as  negative,  and  we  then  have 
_MY-MT' 
^~    M  +  M'    ■ 
The  body  M'  being  supposed  at  rest,  and  impinged  against 
by  the  body  M,  V  will  become  equal   to  zero,  and  the  pre- 
ceding formula  will  reduce  to 

MV 
^~M  +  M'' 
If  the  bodies  have  equal  masses  and  move  in  the  same  direc- 
tion, we  shall  have  M=!M'  ;  and  consequently, 

^=i(V4-V'), 
or,  if  they  move  in  contrary  directions, 

and  when  the  body  M  impinges  upon  an  equal  msiss  M'  at 
rest,  this  expression  reduces  to 

Of  the  Direct  Impact  of  Elastic  Bodies. 

555.  We  will  first  consider  the  circumstances  of  motion 
when  an  elastic  spherical  body  impinges  upon  an  immoveable 
plane  AB  {F^g.  197)  in  a  direction  perpendicular  to  the  sur- 
face of  the  plane.  At  the  instant  when  the  body  comes  in 
contact  with  the  plane,  it  will  begin  to  experience  a  com- 
pression in  the  direction  of  the  diameter  ED,  the  point  D 
being  caused  to  approach  the  centre  of  the  sphere.     This 


COLLISION  OF  ELASTIC  BODIES.  295 

eôect  will  continue  until  the  velocity  of  the  sphere  is  entirely- 
destroyed  ;  then,  in  virtue  of  the  elasticity  possessed  by  the 
body,  an  equal  velocity  will  be  generated  in  an  opposite  direc- 
tion, the  body  at  the  same  time  resuming  its  original  figure. 
Hence,  the  body  will  recoil  with  a  velocity  precisely  equal  to 
that  vnth  which  it  impinged  upon  the  plane. 

550.  Let  us  next  consider  the  impact  of  two  elastic  bodies 
M  and  M'  {Fig.  196),  which  move  in  the  same  direction 
from  A  towards  C,  with  velocities  represented  by  Y  and  V. 
Tliat  an  impact  may  be  possible,  it  is  necessary  that  the 
velocity  of  M  should  exceed  that  of  M'.  When  the  body  M 
overtakes  M',  a  mutual  compression  will  commence,  and  will 
continue  until  tlie  bodies  have  acquired  a  common  velocity  ; 
so  that  a  material  point  D  of  the  body  M  {Pig.  198),  which, 
in  virtue  of  the  velocity  V,  would  have  described  the  lino  DE, 
being  retarded  in  its  motion  by  the  eftect  of  the  compression, 
will,  instead  of  having  reached  the  point  E  at  the  instant  of 
maximum  of  compression,  have  only  arrived  at  a  point  F  : 
then  the  force  of  restitution,  beginning  to  act  upon  the  mate- 
rial point,  will  communicate  to  it  a  velocity  in  a  direction 
opposite  to  that  of  the  motion,  equal  to  that  which  it  has  lost 
by  the  compression,  and  which  would  transfer  it  to  the  ex- 
tremity G  of  a  line  FG=EF,  whilst  the  body  is  resuming  its 
original  figure. 

The  velocity  of  the  body  being  common  to  all  its  points, 
(Art.  443),  if  we  represent  this  velocity  before  impact  by  DE, 
it  may  be  represented  after  impact  by 

DE-GE=:DE-2FE. 

557.  To  express  these  conditions  analytically,  let  u  repre- 
sent the  velocity  common  to  all  the  particles  of  the  two 
bodies  at  the  moment  of  maximum  compression.  At  this 
instant,  the  bodies  may  be  regarded  as  unelastic,  and  the 
velocity  m  will  therefore  be  given  by  the  formula 
MV+MT- 

''=    M  +  M-     (^^^)- 

The  velocity  lost  by  the  body  M  during  the  compression, 
being  equal  to  the  velocity  V  diminished  by  that  which 
remains  at  the  instant  of  greatest  compression,  it  will  be  ex- 
pressed by  V— ?/.     Such  will  be  the  velocity   lost  at  the 


296 


DYNAMICS. 


moment  of  greatest  compression,  but  the  force  of  elasticity, 
tending  to  restore  the  figures  of  the  bodies,  will  cause  the 
body  M  to  sustain  an  additional  loss  of  velocity,  represented 
by  V — u  ;  tluis,  the  total  loss  of  velocity  experienced  by  M 
will  be  expressed  by  2(V— ?i).  Let  v  denote  the  velocity  of 
the  mass  M  after  the  impact  ;  we  shall  have 

or,  by  reduction, 

v=2m— V (315); 

The  body  M',  at  the  instant  of  greatest  compression,  may 
likewise  be  regarded  as  unelastic,  and  will  then  have  gained 
a  velocity  expressed  by  ii — V  :  for  the  velocity  gained  is  evi- 
dently equal  to  the  velocity  u  which  the  body  has  at  this 
instant,  diminished  by  the  original  velocity  V.  The  force  of 
restitution, being  then  exerted,  will  cause  the  body  to  gain  the 
additional  velocity  u — V  ;  whence,  the  entire  gain  of  velocity 
by  M'  will  be  equal  to  2(z^— V),  and  the  velocity  of  M',  after 
collision,  will  therefore  be  expressed  by 

Representing  this  velocity  by  v'y  we  have 

v'=2u—T (316). 

By  substituting  in  equations  (315)  and  (316)  the  value  of 
u  given  by  (314),  we  find 

2(MV+MT0  2(MV  +  MT0 

M  +  M'  V:    ^-      M+M'  ' 

from  which,  by  reduction,  we  obtain 

_V(M-M0+2MT'      ,_r(M^-M)+2MV  ,g,„ 

"^ M+W'       '  '" MTW        ^^'^^• 

If  M=M',  we  shall  have 

v=V',    v'^V (318). 

These  equations  indicate,  that  when  the  masses  are  equal, 
the  impact  will  cause  them  to  exchange  velocities. 

558.  If  the  bodies  move  in  opposite  directions,  the  velocity 
V  may  be  regarded  as  negative  in  the  preceding  formulas, 
which  then  become 

V(M-M')-2M'V'      ,    V'(M— M')+2MV  .„.„. 

""^  M+W         '   '" M+W ^^^^^' 


COLLISION   OF    BODIES.  297 

559.  The  bodies  being  supposed  equal  in  mass,  and  moving" 
in  opposite  directions,  we  make  M=M'  in  equations  (319), 
which  are  thus  reduced  to 

v=-Y',    v'=\ (320). 

Hence  we  conchide  that  the  bodies  will  recoil,  having  ex- 
changed velocities. 

560.  When  the  bodies  impinge  in  opposite  directions,  with 
equal  velocities,  the  masses  of  the  two  being  unequal,  we 
make  V'=V  in  equations  (319),  and  thus  obtain 

_Y(M-3iVr)       , _  V(3M-M') 
^       M+ivr    '  '"~    M+M     ' 
In  this  case,  the  motion  of  M  will  be  entirely  destroyed  by 
the  impact,  if  its  mass  be  supposed  triple  that  of  M'  ;  for 
when  M=3M',  the  first  equation  reduces  to  v=0  :  the  same 
supposition  gives  'y'=2V. 

561.  Lastly,  the  body  M'  being  supposed  at  rest,  and 
impinged  against  by  an  equal  body  M,  we  make  M=M',  and 
V'=.0,  in  equations  (317),  and  we  thus  have 

v=0,     v'=y: 
hence,  the  body  M  will  be  brought  to  rest,  and  M'  will  acquire 
its  entire  velocity. 

Of  the  Preservation  of  the  Motion  of  the  Centre  of  Gravity 
in  the  Impact  of  Bodies, 

562.  Let  the  two  bodies  M  and  M'  be  supposed  to  have 
arrived  at  the  positions  B  and  C  {Fig.  199),  immediately  before 
impinging  upon  each  other  ;  and  let  S  and  S'  represent  their 
distances  from  the  point  A,  and  X  the  distance  of  their  com- 
mon centre  of  gravity  from  the  same  point.  From  the  known 
property  of  the  centre  of  gravity,  we  shall  have 

(M-1-M')X=MS+M'S'; 
and  since  the  distances  X,  S,  and  S'  vary  with  the  time  t,  we 
shall  obtain,  by  differentiating  with  reference  to  t, 

(M+M')^=M§+M'^'. 
at  at  ai 


298  DYNAMICS. 

The  differential  coefficients  -^  and  -r-  represent  the  velo- 

dt  dt      ^ 

cities  of  the  bodies  M  and  M'  at  the  instant  when  they  have 

arrived  at  the  points  B  and  C,  the  distances  of  which  from  the 

point  A  are  represented  by  S  and  S'  respectively.     Let  these 

velocities  be  denoted  by  V  and  V',  and  that  of  the  centre  of 

gravity  by  W=-7-  :  we  shall  obtain,  by  substitution, 

W="+M-V- 

M  +  M'  ^      ' 

Such  is  the  expression  for  the  velocity  of  the  common  centre 
of  gravity  before  the  impact  :  but  immediately  after  the  im- 
pact, the  bodies,  being  found  ai  the  points  B'  and  C,  will  have 
experienced  a  change  in  their  velocities,  and  it  is  required  to 
determine  what  effect  has  been  produced  upon,  the  velocity 
of  their  centre  of  gravity.  Let  w  denote  the  velocity  of  the 
common  centre  of  gravity  after  impact,  and  x  its  distance  from 
the  point  A,  in  the  new  positions  of  the  bodies  ;  the  distances 
of  the  bodies  from  A  being  represented  by  s  and  s'  respectively, 
and  their  velocities  by  U  and  U',  we  shall  have,  as  above, 

(M+M>=M5+MV  : 
and  by  differentiating  with  reference  to  t,  we  find 

(M  +  M')^=M^4-M'^'. 
^  '  dt         dt  dt 

Replacing  — -,  -—,  and  -r-  by  their  respective  values  w^  U,  and 
dt    dt  dt 

U',  there  results 

MU+iAfU- 

'^~    M+M'      ^^^^^' 

563.  Two  different  cases  may  now  be  presented  for  exami- 
nation ;  viz.  the  bodies  may  be  elastic,  or  they  may  be  un- 
elastic  ;  when  they  are  unelastic,  we  have 

U=w=U'; 
whence. 

.  M+M' 

M  +  M' 


COLLISION    OF    BODIES.  299 

But  it  has  been  shown  (Art.  553),  that  the  velocity  lo  common 
to  the  two  bodies  after  the  impact  will  be  equal  to 
MV+MV 
M+M'     ' 
this  velocity  being  precisely  equal  to  the  velocity  W,  it  fol- 
lows that  we  shall  have  w  =  W  ]  or,  the  velocity  of  the  com- 
mon centre  of  gravity  of  two  unelastic  bodies  is  not  affected 
by  their  im,pact. 

564.  When  the  bodies  are  elastic,  their  velocities  after  im- 
pact will  be  expressed  (Art.  557)  by  2w— V,  and  2m — V. 

Substituting  these  values'  of  U  and  U'  in  equation  (322), 
we  find 

m{2u—Y)-[-M'(2u—Y') 
^~  M-fM'  ' 

or,  by  reduction, 

„      MV  +  M'V 
^=^^^-    M  +  M-    '■ 
replacing  the  second  term  of  the  second  member  by  its  value 
u,  there  will  result 

w=7i  ; 
or, 

MV-fM'V 

'w:=i : 

M+M' 
and  eliminating  the  second   member  of  this  equation  by 
means  of  equation  (321),  we  find 

hence  we  conclude,  that  in  the  im,pact  of  elastic  bodies,  as  in 
that  of  unelastic  bodies,  the  velocity  of  the  centre  of  gravity 
is  the  same  before  and  after  impact. 

Of  the  Preservatioîi  of  living  Forces  in  the  Impact  of 
Elastic  Bodies — Relative  Velocity  before  and  after  Im- 
pact— Loss  of  living  Force  in  the  Collision  of  Unelastic 
Bodies. 

565.  The  principle  of  the  preservation  of  living  forces  in 
the  collision  of  elastic  bodies  may  be  enunciated  as  follows  : 

Wlien  two  elastic  bodies  impinge  on  each  other,  the  sunt  of 
their  living  forces  is  the  same  before  and  after  imj'act. 


300  DYNAMICS. 

Let  V  and  V  represent  the  velocities  of  the  bodies  before 
colHsion,  and  v  and  v'  their  velocities  after  collision  ;  the  sum 
of  the  living  forces  before  the  impact  will  be  expressed  by 
MV^+M'V'2  ;  and  it  is  required  to  prove  that  this  sum  is 
equal  to  Mv^  +M'î;'=,  the  sum  of  the  living  forces  after  the 
impact. 

It  has  been  shown  (Art.  557),  that  the  velocities  v  and  v', 
after  impact,  are  given  by  the  equations 

v==2u—Y,     v'=2u-\", 
hence, 

mv^-\-M'v'^=M(2ii—Yy+M'{2u--Y'y  ; 

and  by  performing  the  operations  mdicated  in  the  second 
member,  we  have 

Mi;=^  +MV2  =MV2  +M'Y'2 
+4(Mm2+M'm''— MVî^-M'V'm) (323)  : 

but  the  terms  included  within  the  brackets  mutually  destroy 
each  other,  in  consequence  of  the  relation  (314), 

_MV+]vrv^ 

''        M+M'    ' 
for,  by  clearing  the  denominator,  aiid  multiplying  by  u,  we 
find 

M?^^  i-Wu"'  ^MY  If -{-M'Y '71  ; 
consequently,  the  equation  (323)  will  reduce  to 

Mv^  +MV2  =MV2  +M'V'2. 
This  equation  may  be  written  under  the  form 
Mv^  -f  M'î;'^  —MY'  — M'V'2  =0  ; 
from  which  we  conclude  that  when  elastic  bodies  impinge 
on  each  other,  the  difference  between  the  sums  of  their  living 
forces  before  and  after  impact,  will  be  equal  to  zero. 

566.  The  relative  velocity  of  the  two  bodies  is  the  velocity 
with  which  they  approach  towards,  or  recede  from,  each 
other  ;  and  another  remarkable  property  of  elastic  bodies  con- 
sists in  the  equality  of  their  relative  velocities  before  and  after 
impact.     This  may  be  proved  by  subtracting  the  equations 

v=2u—Y,    v'=2u—Y'] 
from  which  we  obtain 

v-v'=-{Y-Y')  ; 
hence  v'  exceeds  v  by  the  same  quantity  that  V  surpasses  V'  ; 


PRINCIPLE   OF    d'aLEMBERT.  301 

and  the  bodies  will  therefore  separate  after  impact,  with  a 
velocity  precisely  equal  to  that  with  which  they  approached. 

567.  In  the  collision  of  unela^tic  bodies,  the  difference  be- 
tween the  sums  of  the  living  forces  before  and  after  impact 
will  not  be  equal  to  zero  ;  but  it  will  be  equal  to  the  sum  of 
the  living  forces  of  the  bodies  when  moving  with  the  veloci- 
ties lost  or  gained. 

This  theorem  is  due  to  Carnot,  and  may  be  demonstrated 
in  the  following  manner  : 

The  velocities  lost  and  gained  by  M  and  M'  respectively, 
being  equal  to  V— ?«,  m  — V,  if  the  masses  were  moved  with 
these  velocities,  their  living  forces  would  be  expressed  by 

M{Y-uy,    M'(m-V')=  ; 
performing  the  operations  indicated,  we  shall  have 
M(V  -uy-  +M'(u-Y')"~  = 

MV-^+M'V'2  +  (M+M')w2-2m(MV+M'V') (324)  ; 

eliminating  MV-fM'V,  by  means  of  the  equation 
_MV+M'V' 
"        xM+M'    ' 
the  second  member  of  equation  (324)  will  reduce  to 

MV^' +M'V'2— (M+M')«% 
and  we  shall  therefore  have 

M(V-tiy  +M'(^i-V')==MV2+M'V'2-(M+M')M^  ; 
hence  the  truth  of  the  theorem  enunciated  becomes  apparent. 

Principle  of  U Alembert. 

568.  When  the  several  bodies  which  compose  a  system  are 
connected  together  in  any  manner,  and  subjected  to  the  action 
of  different  forces,  this  connexion  will  in  general  prevent 
each  body  from  taking  the  motion  which  would  have  been 
communicated  to  it  if  the  connexion  had  not  existed.  For 
example,  if  several  material  points  M,  M',  M",  &c.  [Fig.  200) 
be  attached  to  an  inflexible  right  line  AL,  moveable  about  the 
point  A,  it  is  evident  that  these  points,  being  unable  to  move 
except  with  the  line  AL,  will,  when  acted  on  by  the  force  of 
gravity,  oscillate  together  about  the  point  A,  describing  arcs 

26 


302 


DYNAMICS. 


proportional  to  their  distances  from  A,  and  will  at  the  end  of 
a  certain  time  be  brought  into  the  positions  K,  K',  K",  &.C.; 
whereas,  if  the  points  were  unconnected,  being  merely  attached 
to  the  point  A,  they  would,  from  the  principles  of  the  simple 
pendulum,  explained  in  Art.  471,  oscillate  in  very  unequal 
times,  depending  on  their  distances  from  the  point  A.  More- 
over, if  we  resolve  each  of  the  several  forces  which  are  exerted 
in  vertical  directions  upon  the  points  M,  M',  M",  &.C.,  into 
two  components,  one  of  which  shall  act  along  the  line  AL, 
and  the  other  in  a  direction  perpendicular  to  this  line  ;  the 
latter  component  will  alone  tend  to  communicate  motion  to 
the  point  ;  and  since  the  several  perpendicular  components, 
exerted  on  the  different  points,  will  be  equal  to  each  other, 
they  would  communicate  in  the  instant  of  time  c?^  equal  veloci- 
ties to  the  points  M,  M',  M"  <fcc.,  if  these  points  were  uncon- 
nected. But  in  consequence  of  their  connexion,  the  veloci- 
ties assumed  are  evidently  proportional  to  their  distances  from 
the  point  A. 

569.  It  thus  appears,  that  the  effective  velocities  assumed 
by  the  several  parts  of  the  system  differ  from  the  velocities 
impressed,  and  hence  the  circumstances  of  the  motion  can 
only  be  discovered  when  we  have  succeeded  in  expressing  the 
effective  velocities  in  functions  of  the  velocities  impressed. 
This  object  is  readily  accomplished  with  the  assistance  of  a 
dynamical  principle  first  employed  by  D'Alembert. 

570.  Let  v,  v\  v",  (fee.  represent  the  velocities  which  Avould 
be  impressed  by  certain  forces  on  the  bodies  M,  M',  M",  &c., 
if  they  were  perfectly  free,  and  u,  u',  ii",  &c,  the  velocities 
assumed  by  these  bodies  in  consequence  of  their  connexion. 
The  velocity  v  being  resolved  into  two  components,  one  of 
these  components  may  be  assumed  arbitrarily,  and  the  second 
will  then  become  determinate.  Let  the  effective  velocity  ii 
be  assumed  as  the  arbitrary  component  of  the  impressed 
velocity  v,  and  denote  the  other  component  by  U.  Making  a 
similar  decomposition  of  the  other  velocities  v',v",  &c.,  we  have 

u  and  U  for  the  components  of  v, 
u'  and  U'  for  those  of  v', 
u"  and  U"  for  those  of  v", 
(fee.  &.C.  (fee.  ; 


PRINCIPLE    OF    d'aLEMBERT.  303 

and  the  quantities  of  motion  impressed  upon  the  system, 
which  are  Mv,  M'v',  M"v",  &c.,  will  become,  after  the  de- 
composition, 

Mu,  MV,  M"u",  (fee, 

MU,  M'U',  M"U",  &c. 

But,  in  consequence  of  the  connexion  of  the  different  parts  of 
the  system,  these  quantities  of  motion  will  be  reduced  to 

Mu,  M'«',  M'V,  (fee.  ; 
hence,  it  is  necessary  that  the  quantities  of  motion  MU,  M'U', 
M"U",  (fee.  should  destroy  each  other,  or  should  produce  an 
equilibrium.  For,  if  it  were  otherwise,  we  might  combine  the 
resultant  of  the  quantities  of  motion  MU,  M'U',  M"U",  cfec. 
with  the  quantities  of  motion  Mu,  M'u',  M"w",  (fee.  ;  thus  the 
effective  velocities  of  the  several  parts  of  the  system  would 
no  longer  be  represented  by  u,  u',  u",  (fee,  which  is  contrary 
to  the  hypothesis. 

571.  It  may  be  observed  that  the  products  MU,  M'U', 
M"U",  (fee.  express  the  quantities  of  motion  due  to  the  veloci- 
ties  lost  or  gained  by  the  several  bodies.  For  the  velocity  v 
may  be  replaced  by  its  two  components  u  and  U  ;  the  former 
of  which  expresses  the  effective  velocity  of  the  body  M,  and 
the  latter  represents  that  velocity  which,  combined  with  ii, 
would  produce  the  impressed  velocity.  Thus,  U  is  a  velocity 
introduced  or  destroyed  in  the  system  by  the  connexion  of  its 
parts. 

The  general  principle  may  therefore  be  enunciated  in  the 
following  manner  :  It  is  necessary  that  the  quantities  of 
motion  due  to  the  velocities  lost  or  gained  should  he  such  as 
would  maintain  the  system  hi  equilibrio. 

572.  It  has  been  remarked  that  the  quantity  of  motion  Mv 
may  be  resolved  into  the  two  components  Mu  and  MU  ;  and 
since  an  equilibrium  will  always  subsist  between  three  forces, 
one  of  which  is  equal  and  directly  opposed  to  the  resultant 
of  the  other  two,  it  follows  that  the  forces  represented  by  Mu 
and  MU  will  sustain  in  equilibrio  a  force  equal  and  opposite 
to  Mv  ;  and  consequently,  that  the  force  Mv  will  sustain  in 
equilibrio  two  forces  which  are  respectively  equal  and  oppo- 
site to  Mu  and  MU. 


304  DYNAMICS. 

The  same  remarks  being  applicable  to  the  other  forces,  it 
appears  that  the  forces  Mr,  M'v',  M"v",  &c.  will  sustain  in 
equilibrio  two  systems  offerees  which  are  equal  and  directly 
opposed  to  the  forces 

Mu,  M'w',  M"u",  &c., 
MU,  MU',  M"U",  &c. 
But  the  forces  MU,  M'U',  M"U",  (fcc.  destroy  each  other  ;  and 
hence  we  obtain  a  second  enunciation  of  the  principle  of 
D'Alembert,  viz.  ;  An  equilibrium  will  subsist  between  the 
quantities  of  motion  Mî;,  JM'v',  M"v",  cj'c.  impressed  upon  the 
several  bodies,  and  the  effective  quantities  of  motion  Mtf, 
M'?<',  M'w",  ^'c,  tJie  latter  being  applied  in  dii^ections  con- 
trary to  those  of  the  motions  actually  asstimed. 

573.  This  principle  is  equally  true,  whether  the  velocities 
V,  v',  v",  &yC.  are  finite  velocities,  acquired  by  the  masses  M, 
M',  M",  &c.  during  a  finite  time,  or  communicated  instanta- 
neously by  forces  of  impulsion  ;  or,  when  these  velocities  are 
infinitely  small,  being  generated  by  incessant  forces  ;  or, 
finally,  when  some  of  these  velocities  are  finite,  and  some  of 
them  infinitely  small. 

574.  To  apply  this  principle,  let  us  consider  the  impact  of 
two  unelastic  bodies  M  and  M',  which  move  in  the  same 
direction.  Let  v  and  v'  represent  their  velocities  before  im- 
pact, and  71  the  common  velo  'ty  after  impact.  The  velocity 
lost  by  M  being  equal  to  its  original  velocity  diminished  by  that 
which  remains  after  collision,  it  will  be  expressed  by  ?;—?«: 
in  like  manner,  the  velocity  lost  by  M'  will  be  expressed  by 
!>' — 11.  The  quantities  of  motion  due  to  these  velocities  being 
such,  by  the  principle  of  D'Alembert,  as  to  produce  an  equi- 
libriinn,  we  shall  have 

M{v—2i)  +  M'{v' -zi)=0  ; 
whence  we  deduce  for  the  velocity  after  impact, 
_Mv  +  M'v' 
''~  M+M'  • 

When  the  bodies  move  in  opposite  directions,  v'  will  become 
negative. 

575.  As  a  second  example,  let  it  be  required  to  determine 
the  circumstances  of  motion  of  two  bodies  M  and  M',  which 


PRINCIPLE    OF    d'aLEMBERT.  ;  305 

rest  on  two  inclined  planes  AB  and  AC  {Fig.  201)  having  a 
common  altitude,  and  are  connected  by  a  thread  MEM',  pass- 
ing over  a  fixed  pulley. 

If  the  vertical  line  M^,  drawn  through  the  centre  of  gravity 
of  the  body  M,  be  supposed  to  represent  the  intensity  of  the 
force  of  gravity  ;  the  component  of  the  force  in  the  direction 
of  the  plane  will  be  represented  by  MR  ;  this  component  will 
alone  tend  to  urge  the  body  down  the  plane  :  its  value  will 
be  expressed  by 

AD 
§•  Xcos  RMg-^^ .  cos  BAD=^— — . 

AB 

In  like  manner,  the  component  of  gravity,  which  tends  to 
cause  the  descent  of  the  body  M'  on  the  plane  AC,  will  be 

expressed  by  g-j-^- 
AC 

Let  the  lines  AD,  AB,  and  AC  be  denoted  by  h,  I,  and  V 
respectively  ;  the  incessant  forces  exerted  upon  the  bodies 
will  then  be 

gh        J  gh 

T'  ""■*  T- 

But  if  we  suppose  the  motion  to  take  place  in  the  direction 
M'EM,  and  the  velocities  to  be  reckoned  as  positive  in  this 

direction,  the  force  j^,  which  is  opposed  to  the  motion,  must 

be  regarded  as  negative  ;  and  the  incessant  forces  will  there- 
fore be  expressed  by 

it,  and   _f . 

The  general  expression  for  the  value  of  an  incessant  force 
being 


dv 

<P  = 

we  have 


^=^^' 


dv=ç>dt  : 
hence,  the  velocities  imparted  to  the  bodies  in  the  time  dU 
when  they  are  unconnected,  will  be  expressed  by 

u 


306  DYNAMICS. 

and  the  quantities  of  motion  due  to  these  velocities  will  be 

Ug!^di,      -WffydL 

But  the  bodies  being  supposed  connected  by  a  thread  of  inva- 
riable length,  if  M  should  descend  through  any  distance  on 
the  plane  AB,  M'  will  necessarily  ascend  through  an  equal 
distance  on  the  plane  AC  ;  or,  in  other  words,  the  velocities 
of  the  bodies  at  any  instant  will  be  equal  to  each  other.  De- 
noting by  V  their  common  velocity  at  the  end  of  the  time  t, 
the  eifective  velocities  communicated  to  them  in  the  succeed- 
ing instant  dt,  will  be  expressed  by  dv,  and  the  effective 
quantity  of  motion  imparted  in  the  same  time,  will  there- 
fore be 

(M.-\-M')dv. 

By  the  principle  of  D'Alembert,  this  quantity  of  motion  when 
applied  in  a  contrary  direction,  wil  produce  an  equilibrium 
with  the  quantities  of  motion  impressed  on  the  bodies  :  hence, 
the  sum  of  these  quantities  of  motion  will  be  equal  to  zero,  or 

-{M+M)dv+Mg^dt-^^dt=-0 (325)  : 

from  which  we  deduce 

and  by  integration, 

■      "=4+^-^'+° (^'^"'^ 

or,  if  we  denote  by  G  the  coefficient  of  ;,  we  shall  have 

v=Gt-\-0 (326). 

Let  X  represent  the  distance  OK  of  the  body  M  from  the 
point  O,  the  origin  of  the  spaces,  at  the  end  of  the  time  t  ; 
the  general  expression  for  the  velocity  gives 

dx 


and  therefore, 


'''  'dt 


at 


PRINCIPLE    OF    d'aLEMBERT.  307 

from  which,  by  integration,  we  obtain 

ar=iG^2+o^  +  C' (327). 

The  formulas  (326)  and  (327)  indicate  that  the  circumstances 
of  motion  in  this  system  are  precisely  similar  to  those  which 
attend  the  fall  of  heavy  bodies  ;  the  only  difference  consisting 
in  the  value  of  the  incessant  force,  which  in  the  latter  case  is 
denoted  by  g,  and  in  the  former  by  G. 

576.  If  the  planes  AB  and  AC  be  supposed  to  become  ver- 
tical, the  case  will  be  reduced  to  that  of  two  weights  con- 
nected by  a  cord  which  passes  over  a  fixed  pulley  :  the  quan- 
tities A,  Z,  and  I'  are  then  equal,  and  the  equations  (325  a)  and 
(327)  may  then  be  reduced  to 

M— M'  M— M' 

''=ffw^^'+'''  "=ra''*^''+'''+''' •  •  •  <'''''^- 

577.  These  formulas  will  serve  to  explain  the  principle  of 
Atwood's  machine,  which  is  employed  for  the  verification  of 
the  laws  of  constant  forces. 

This  machine  consists  essentially  of,  1°,  A  fixed  pulley, 
over  which  passes  a  very  fine  flexible  thread,  having  its  ex- 
tremities attached  to  two  equal  brass  basins  ;  2°.  A  vertical 
graduated  scale  with  a  moveabls  stage  to  maxk  the  space 
passed  over  by  the  descending  basin  ;  and,  3°.  A  seconds 
pendulum,  by  means  of  which  the  time  of  descent  may  be 
accurately  observed. 

When  the  two  basins  are  loaded  with  equal  weights,  they 
will  sustain  each  other  in  equilibrio  ;  but  if  an  addition  be 
made  to  either,  it  will  immediately  preponderate,  aîid  will 
produce  a  motion  uniformly  varied.  Moreover,  by  rendering 
the  difference  M— M'  of  the  weights  M  and  M'  attached  to 
the  extremities  of  the  thread,  very  small  in  comparison  with 
their  sum  M +  M',  the  space  described  and  the  velocity  ac- 
quired in  a  given  time  which  result  from  equations  (327  a) 
may  likewise  be  rendered  small,  and  the  observations  will 
thus  become  susceptible  of  great  accuracy. 

For  the  purpose  of  observing  the  velocity  acquired  at  the 
end  of  any  time,  we  give  to  the  additional  weight  placed  in 
the  descending  basin  the  form  of  a  flat  bar,  and  the  basin 
being  allowed  to  pass  through  a  sliding  ring  attached  to  the 

U2 


308  DYNAMICS. 

vertical  scale,  the  bar  may  be  removed  at  any  instant  during 
the  descent.  The  equality  of  the  weights  in  the  two  basins 
being  restored  by  the  removal  of  the  bar,  the  motion  becomes 
uniform  with  the  velocity  acquired  at  the  instant  when  the 
bar  was  removed. 

By  comparing  the  spaces  described,  the  velocities  acquired, 
and  the  times  elapsed,  we  find  that  when  the  basins  move 
from  rest  under  the  influence  of  a  constant  force,  the  velocities 
are  constantly  ])roj)ortional  to  the  times,  and  that  the  spaces 
are  proportional  to  the  squares  of  the  times. 

578.  For  a  third  example,  let  it  be  required  to  investigate 
the  circumstances  of  motion  of  two  weights  M  and  M',  which 
are  attached  to  cords  passing  around  the  respective  circum- 
ferences of  a  wheel  and  of  its  axle. 

If  we  suppose  the  body  M  to  prevail,  and  reckon  the  veloci- 
ties positive  in  the  direction  of  its  motion,  the  force  of  grav- 
ity will  impress  upon  the  bodies  M  and  M',  in  the  instant  dt^ 
which  succeeds  the  time  t,  the  velocities  gdt  and  — gdt  ;  and 
the  quantities  of  motion  impressed  will  therefore  be 

Mgdt,  and  —Wgdt. 
But  if  V  and  v'  represent  the  velocities  of  M  and  M'  at  the  ex- 
piration of  the  time  t,  the  effective  velocities  communicated 
in  the  succeeding  instant  dt  will  be  expressed  by  dv  and  dv'. 
Thus,  denoting  by  R  and  r  the  radii  of  the  wheel  and  axle, 
we  shall  have 

Masses.       Impressed  velocities.       Effective  velocities.       Distances  from  the  axis. 

M   .  ...  gdt dv R, 

M'  .  .  .  —gdt dv' r. 

The  effective  quantities  of  motion,  being  applied  in  directions 
contrary  to  those  of  the  motions  assumed,  will  sustain  in  equi- 
librio  the  quantities  of  motion  impressed  ;  and  since  the  equi- 
librium is  maintained  through  the  intervention  of  the  wheel 
and  axle,  it  is  necessary  that  the  sum  of  the  moments  with 
reference  to  the  axis  should  be  equal  to  zero  :  hence,  we 
obtain 

WRgdt—Wrgdt-MRdv—'m:rdv'=() (328). 

This  equation  containing  the  two  unknown  quantities  v  and 
r',  it  will  be  necessary  to  discover  a  second  relation  between 


UNIFORM    MOTION   ABOUT    AN    AXIS.  309 

them.  For  this  purpose,  we  remark  that  the  velocities  v 
and  v'  bear  to  each  other  the  constant  ratio  of  R  :  r  ;  thus,  we 
have 

V  :  v'  ::B.  :  r  ] 
or, 

r 


and  by  differentiating, 


V  —V- 

R 


T 

dv'=—dv 
R 


substituting  this  value  in  equation  (328),  we  find 
MRgdt—M'rgdt—MKdv-M'^dv=0  ; 

or,  by  reduction  and  transposition, 

MR'dv-^-M'r^dv^MR^gdt-M'Rrgdt; 
whence, 

,       MR2-M'Rr    ,, 

^^^=mrm:mv^^^'' 

Denoting  by  K  the  constant  coefficient  of  df,  this  equation 
becomes 

dv=K.dt  ; 
and  by  integration, 

Replacing  v  by  its  value  — -,  and  performing  a  second  integra- 
tion, we  find 

These  results  indicate  that  the  motion  is  uniformly  varied, 
the  circumstances  of  the  motion  being  similar  to  those  of  a 
body  falling  under  the  influence  of  the  force  of  gravity. 

Of  the  Motion  of  a  Body  about  a  Fixed  Axis. 

579.  When  an  impulse  is  applied  to  a  system  of  material 
points  connected  together  in  an  invariable  manner,  and  sub- 
jected to  the  condition  of  turning  about  a  fixed  axis,  which 
we  will  suppose  to  pass  through  the  point  A  {Fig.  202),  per- 
pendicular to  the  plane  of  the  figure,  the  several  particles  tw, 


310 


DYNAMICS. 


m\  m",  &c.  will  describe  circles  won,  rr^o'i}!^  m"o"n",  <fcc.,  the 
planes  of  which  will  be  parallel  to  each  other,  and  perpen- 
dicular to  the  fixed  axis  ;  and  the  arcs  described  by  the  several 
points  in  the  same  time  will  contain  the  same  number  of 
degrees.  These  arcs  being  proportional  to  their  radii,  the 
velocities  of  the  several  particles  will  be  in  the  same  propor- 
tion ;  so  that  if  we  denote  by  a  the  velocity  of  the  particle  e, 
whose  distance  eA  from  the  axis  of  rotation  is  equal  to  unity, 
the  velocities  of  the  particles  w,  m.\  m",  <fcc.,  at  the  distances 
r,  r',  r",  &c.  from  the  fixed  axis,  will  be  expressed  by  r*,  r'«, 
r%  &c.  Thus,  the  efiective  quantities  of  motion  of  the  dif- 
ferent particles  Avill  be  represented  by 

mro},  m'r'oj,  'm"r"u,  &c. 

Let  V,  v',  v",  &c.  be  the  velocities  impressed  :  the  correspond- 
ing quantities  of  motion  will  be  expressed  by  niv,  m'v',  ni!'v'\ 
&.C.  It  will  therefore  be  necessary,  according  to  the  second 
enunciation  of  the  principle  of  D'Alembert,  that  an  equilib- 
rium should  subsist  between  the  forces  mv^  oyi'v',  'm"v",  &c., 
and  — mro)^  — ni'r'u^  — 'm"r"o),  &c. 

To  establish  the  conditions  of  equilibrium  between  these 
forces,  we  will  first  consider  the  force  niv,  and  represent  it  by 
w/ja  portion  of  its  line  of  direction  :  from  the  point/ let  the 
perpendicular //i  be  demitted  upon  the  plane  of  the  section 
om/2,and  denote  by  <p  the  angle /?7iA,  formed  by /w  with  this 
plane  ;  by  constructing  the  rectangle  hh',  the  force  mv  may 
be  resolved  into  the  two  components 

mh'  =  (mv) .  sin  <p,  parallel  to  the  fixed  axis, 
mh  =  (mv) .  cos  <p,  situated  in  the  plane  om?i. 

The  first  of  these  components  will  have  no  tendency  to  turn 
the  system  about  the  fixed  axis  ;  but  the  second  will  produce 
its  entire  effect  in  communicating  a  motion  of  rotation. 

Tf  we  represent  in  like  manner  by  <z>',  p",  &.c.  the  angles 
formed  by  the  directions  of  the  forces  m'v',  vi'v",  &c.  with 
the  planes  o'm'?i',  o"m"ii",  (fee,  the  quantities  of  motion  im- 
pressed will  become 

mv  cos  (p,     m!v'  cos  <p\     m"v"  cos  <p",  (fee. 

These  quantities  of  motion,  as  well  as  the  quantities  -  ????>, 


UNIFORM    MOTION    ABOUT    AN    AXIS.  311 

—m'r'u,  —}7i"r"u,  &c.  are  situated  in  planes  perpendicular  to 
the  fixed  axis. 

The  conditions  of  equilibrium  between  these  forces  will 
evidently  be  the  same  as  those  which  arise  when  the  forces 
are  situated  in  the  same  plane  ;  if,  therefore,  the  forces  be 
regarded  as  situated  in  the  plane  of  the  figure,  the  conditions 
of  equilibrium  will  require  that  the  sum  of  the  moments  of 
the  forces  which  tend  to  turn  the  system  in  one  direction 
about  the  point  A,  shall  be  equal  to  the  sum  of  the  moments 
of  those  which  tend  to  produce  rotation  in  a  contrary  direc- 
tion ;  or,  that  the  algebraic  sum  of  the  moments  shall  be 
equal  to  zero. 

But  the  quantities  of  motion  — mr<y,  —m'r'a,  —m"r"a>,  &c. 
being  derived  from  the  common  motion  of  the  system,  they 
will  tend  to  turn  it  in  the  same  direction  ;  and  since  these 
motions  take  place  in  the  circumferences  of  the  circles  mno, 
m'u'o',  m"n"o",  &c.,  the  radii  r,  r',  r",  &c.  will  represent  the 
perpendiculars  demitted  from  the  point  A  upon  their  respect- 
ive directions  ;  consequently,  the  sum  of  the  moments  of  the 
effective  quantities  of  motion,  when  applied  in  opposite  direc- 
tions, will  be  expressed  by 
— mr'^61  —  mr'^u — rn'r'^u  —  &Lc.  =  —  «(mr^-j-TOr'^-f-OT  V'^-i-&c.). 

Let  the  quantity  within  the  brackets  be  denoted  by  :s{mr^)  ; 
the  sum  of  these  quantities  of  motion  will  then  be  repre- 
sented by  — û)2(mr2). 

To  determine  the  value  of  the  sum  of  the  moments  of  the 
impressed  forces, 

mv  .  cos  (p,  mJv' .  cos  <p',  m"v" .  cos  <?",  &c., 
let  Az  {Pig.  203)  represent  the  fixed  axis,  and  ml,  m'l',  m"l", 
&c.  the  forces  qîiv  .  cos  (p,  m'v' .  cos  <?',  ni"v"  .  cos  <p",  <fec.  situated 
in  the  planes  mno,  m'ti'o',  m"n"o",  &c.,  perpendicular  to  the 
fixed  axis  :  from  the  points  A,  A',  A",  &c.,  at  which  the  axis 
intersects  the  perpendicular  planes,  let  the  perpendiculars 
M=p,  A'l'=p',  A"l"=p",  (fcc.  be  demitted  upon  the  directions 
of  the  several  forces  mv  cos  (p,  m'v'  cos  ç>',  m"v"  cos  <f>",  <fcc.  ; 
the  moments  of  these  forces  will  be  expressed  by 

mv  cos  ç> .  p,     m'v'  cos  <p' .  p',     m"v"  cos  p"  .  p". 
The  algebraic  sum  of  these  moments  will  be  expressed  by 


312 


DYNAMICS. 


2(mu  cos  <p  .2^)  ]  and  hence,  by  the  conditions  of  equilibrium 
before  enunciated,  we  shall  have 

2(mv  .  cos  ç>  .  p)  — al.(mr^  )  =0. 
This  equation  gives  the  value  of  the  angular  velocity 
^^Hmv.cosç.p) 

and  the  motion  of  the  body  about  the  fixed  axis  will  there- 
fore be  uniform. 

580.  When  the  forces  mv^  m'v\  m"v",  (fcc.  are  exerted  in 
planes  perpendicular  to  the  axis,  the  angles  <p,  <p',  <p",  &c. 
become  equal  to  zero,  and  we  have 

sin  ^=0,     cos  ^=1, 

sin  ^'=0,    cos  (p'=l, 

sin  ç>"=0,   cos  <p"~l, 

&c.  (fee.  ; 

consequently,  the  equation  (329)  reduces  to 

I.{mvp) 

581.  If  equal  velocities  be  impressed,  in  parallel  directions, 
upon  the  several  particles  m,  m',  m",  (fee,  we  shall  have 

v=v'=v"=(fec. 
and  the  moments  of  the  quantities  of  motion  impressed  will 
become 

mvp  +  ?n'vp' -{•m"vp" -]-ôcc.=v{mp -^7ii'p'  +'m"p"  +  ôcc.)  : 
the  sum  of  these  moments  may  be  represented  by  v^{mp)f 
and  the  equation  (329)  will  be  transformed  into 

.=.!^^ (330). 

Let  a  plane  AK  be  now  drawn  through  the  axis  As  {Fig. 
204),  parallel  to  the  directions  of  the  several  forces  mv,  vi'v', 
7n"v",  (fee.  :  the  perpendiculars  p,  p\  jj",  (fee.  demitted  from  the 
points  A,  A',  A",  (fee.  upon  the  directions  of  these  forces,  are 
evidently  equal  to  the  perpendiculars  mq,  ni'q',  7n"q",  (fee,  let 
fall  from  the  points  ???,  m,',  m'\  (fee.  upon  the  plane  AK.  Let 
q,  q',  q",  (fee,  represent  these  perpendiculars,  and  Q,the  perpen- 
dicular demitted  from  the  centre  of  gravity  of  the  system,  upon 
the  plane  AK  ;  then,  denoting  by  M  the  sum  of  the  particles 


UNIFORM    MOTION    ABOUT    AN    AXIS.  313 

which  compose  the  system,  or  the  entire  mass  of  the  body, 
we  shall  have,  by  the  property  of  the  centre  of  gravity, 

MGi—mq-\-m'q' -\-m,"q" ■\-&LQ,.  ; 
and  since 

p=q,     p'=q',     p"  =  q",  (fec, 
the  preceding  equation  may  be  written 

MGi=mp  -{-fn'jy  -\-'m"p"  -\'(Scc.  =  7:{mp). 
This  value  being  substituted  in  equation  (330),  there  results 
vMGi  .^„.> 

«=: — (OOi). 

582.  It  may  happen  that  the  velocity  v  has  been  impressed 
iipon  only  a  limited  number  of  the  particles  ?n,  m\  m",  &c.  : 
then,  M  will  no  longer  represent  the  entire  mass  of  the  system, 
but  merely  the  sum  of  those  particles  upon  which  the  velocity 
has  been  impressed  ;  and  Q,  will  express  the  perpendicular 
demitted  from  the  centre  of  gravity  of  this  part  of  the  system 
upon  the  plane  AK. 

The  quantity  ^(mr^)  is  called  the  moment  of  inertia  :  the 
method  of  determining  its  value  will  be  explained  in  the  next 
section. 

583.  It  is  frequently  necessary  to  consider  the  effects  pro- 
duced upon  the  fixed  axis  by  the  application  of  an  impulsive 
force  to  any  point  of  the  system.  For  this  purpose,  let  the 
axis  of  rotation  Kz  {Fig.  205),  be  assumed  as  the  axis  of  z, 
and  resolve  the  impulsion  P,  which  is  supposed  to  be  applied 
at  a  point  O,  into  two  components  P'  and  P",  which  shall  be 
respectively  parallel  and  perpendicular  to  the  plane  of  x,  y. 
liCt  the  axis  of  y  be  then  assumed  parallel  to  the  direction 
of  P',  and  denote  the  co-ordinates  of  the  point  O  by  a,  è,  and 
c  :  since  the  force  P  may  be  applied  at  any  point  in  its  line 
of  direction,  we  can  always  suppose  the  point  of  application 
O  to  be  contained  in  the  plane  oi  x^z'.  this  supposition  gives 
6=0. 

Instead  of  regarding  the  axis  as  fixed,  let  such  forces  be 
introduced  as  may  be  necessary  to  retain  it.  These  forces 
will  be  equal,  and  directly  opposed  to  the  impulsions  expe- 
rienced by  the  axis,  and  may  in  general  be  reduced  to  three 
forces  respectively  parallel  to  the  axes  of  ^,  y,  and  z.     Let  X, 

27 


314  DYNAMICS. 

Y,  and  Z  represent  the  impulses  communicated  to  the  axis, 
and  call  AB=«,  AC=/3. 

The  particle  m  will  describe  a  circle  parallel  to  the  plane 
of  X,  y,  and  its  velocity  in  the  direction  of  the  tangent  ml  will 
be  expressed  by  ret  (Art.  579)  :  the  cosines  of  the  angles 
formed  by  this  direction  with  the  axes  of  x  and  y  respectively, 

will  be  —  and  — —  ;  hence,  the  effective  quantity  of  motion  of 
r  r 

the  particle  m  will  be  mrai,  and  its  components  in  the  direction 

of  the  axes  of  x  and  y  will  be  mi/a  and  —mxa  :  the  same 

remarks  apply  to  the  other  particles  m',  m",  m"\  &c. 

But,  by  the  principle  of  D'AIembert,  an  equilibrium  will 

subsist  between  the  effective  forces  and  the  force  P,  the  latter 

being  applied  in  a  contrary  direction  ;  thus,  we  shall  have 

Forces.      Components  parallel  to  axes  of  Co-ordinates  of  points  of  application  parallel  to 

X  y  z  X  y  z 

— P        0       Pcos^     — Psin^.  ...  a  0  c, 

X        X  0  0 0  0 

Y         0  Y  0 0  0  ^, 

Z         0  0  Z 0  0  0, 

niru      myu     — mxu  0 x  y  z, 

m'r'a)     m'y' a  — mx'u  0 x'  y'  z\ 

<fcc.  (fee.  &c. 

The  general  equations  (66)  and  (67),  which  express  the 
conditions  of  equihbrium  of  forces  lying  in  different  planes, 
and  acting  upon  various  points  of  a  body,  may  be  written 
under  the  form 

2(X)=0,  2:(Xy-Y:i')=0, 
2(Y)=0,  2(Zt-Xz)=0, 
s(Z)=0,     s(Y;s-Zy)=0; 

and  when  applied  to  the  system  under  consideration,  will 

give 

X4-*2;(m2/)=0, 

Y  +  P  cos  ^  — «2(m:r)=0, 

Z— Psin^=0; 

«2:(7/w=)  —  P  cos  ç>a=0, 

X«+«2;(my2;)  +  P  sin  ^a=0, 

Y/3+P  cos  <pc  —a'z{?nxz)=0. 


UNIFORIW    MOTION   ABOUT    AN   AXIS.  315 

Let  M  represent  the  mass  of  the  body,  r„  y„  and  z,  the 
co-ordinates  of  its  centre  of  gravity,  and  Mv  the  quantity  of 
motion  which  the  force  P  is  capable  of  communicating  :  these 
six  equations  will  be  reduced  to 

Y=cMx,—mv . cos ^  V (331  a)  ; 

Z=Mv.sin<^  5 

«s(mr2)=My  .cos^  .  a  ^ 

X<«=  — «s(my2;)— Mv  .  sin  p .  a  > (331  b). 

Y^=<u'z(ni.Tz)  — Mv  .  cos  p  .c  j 
From  the  fourth  equation  we  deduce  the  value  of  u,  which 
being  substituted  in  the  first  and  second,  the  values  of  X  and 
Y  become  known  :  the  third  determines  the  value  of  Z,  and 
the  fifth  and  sixth  give  the  co-ordinates  «  and  /3  of  the  points 
B  and  C,  at  which  the  forces  X  and  Y  are  applied.  The 
solution  of  the  problem  is  therefore  complete. 

When  we  wish  to  communicate  the  impulse  P  in  such  a 
manner  that  the  axis  shall  receive  no  shock,  we  make  X,  Y, 
and  Z  equal  to  zero.  This  supposition  reduces  the  equations 
(331  a)  and  (331  b)  to  the  following  forms  : 

ax, = V,       ^{myz) = 0, 

The  third  equation  indicates  that  the  direction  of  the  impulse 
must  be  parallel  to  the  plane  of  x,y;  the  first,  that  the  centre 
of  gravity  of  the  body  must  lie  in  the  plane  of  .r,  z,  perpen- 
dicular to  which  the  impulse  is  applied  ;  the  second  deter- 
mines the  angular  velocity  a  ;  and  the  fourth  and  sixth  make 
known  the  vakies  of  the  co-ordinates  a  and  c  of  the  point  O. 
The  point  O  is  then  called  the  centre  of  percussion^  which 
may  be  defined  to  be  that  point  in  the  plane  passing  through 
the  centre  of  gravity  and  the  axis  of  rotation,  at  which  an 
hnpulse  must  be  applied  in  a  direction  perpendicidar  to  this 
plane,  in  order  that  the  axis  may  receive  no  shock. 

584.  The  equation  l(myz)=0  expresses  a  relation  which 
is  evidently  dependent  on  the  figure  of  the  body  and  tlie 
position  of  the  axis  of  rotation.  This  relation  will  exist  only 
in  particular  cases,  and   it   therefore   follows   that  a  body 


3lB  DYNAMICS. 

retained  by  a  fixed  axis  will  not  necessarily  have  a  centre  of 
percussion. 

585.  The  distance  of  the  centre  of  percussion  from  the 
axis  of  rotation  being  equal  to  the  absciss  AN=:a,  its  value 
will  be 

a= — ^ — _:  =  -> '-. 

586.  Although  the  axis  will  receive  no  impulse  at  the  in- 
stant of  impact,  yet  the  motion  of  rotation  will  immediately 
give  rise  to  centrifugal  forces  which  will  exert  a  pressure 
upon  the  axis. 


Of  the  Moment  of  Inertia. 

587.  The  momert  of  inertia  being  the  sum  of  the  products 
formed  by  multiplying  each  material  point  of  a  system  by 
the  square  of  its  distance  from  a  fixed  axis,  it  has  been  repre- 
sented in  the  preceding  section  by  2(wir").  In  this  expres- 
sion, we  may  replace  the  particle  m  by  dM.,  the  element  of  the 
mass  ;  and  the  moment  of  inertia  will  then  result  from  the 
integration  of  an  expression  of  the  form/r^rfM. 

588.  For  example,  let  it  be  required  to  determine  the 
moment  of  inertia  of  a  material  right  line  CB  {Pig.  206),  with 
reference  to  an  a\is  AZ  perpendicular  to  the  plane  CAB. 

Let  AB=/i  represent  the  perpendicular  demitted  from  the 
point  A  upon  the  right  line,  and  BP=;r  the  distance  of  a 
point  P  assumed  arbitrarily  on  this  line,  from  the  point  B  : 
we  shall  have 

PA2=A2^.'r2. 

This  expression  being  multiplied  by  the  diflerential  of  the 
mass,  the  integral  of  the  product  will  express  the  moment  of 
inertia.  The  volume,  in  the  present  case,  being  a  right  line, 
the  element  of  the  volume  will  be  represented  by  the  infinitely 
small  difference  dx  between  two  consecutive  abscisses  BP=^ 
and  BV=x-\-dx]  and  the  element  of  the  mass  rfM  will 
therefore  be  expressed  by  dx  multiplied  by  the  density  D,  or 
by  Ddx.     Thus,  by  multiplying  h-  -^-x^  by  Ddx^  and  inte- 


MOMENT   OF    INERTIA.  317 

grating,  we  obtain  for  the  expression  of  the  moment  of  inertia 
of  the  right  hne, 

In  the  present  disposition  of  the  figure,  the  integral  should  be 
taken  between  the  limits  of  the  point  B,  where  x=0,  and  the 
point  C,  at  which  x=a]  the  moment  of  inertia  thus  becomes 


('"«+Ï)d- 


In  effecting  this  integration,  we  have  regarded  the  line  as 
homogeneous,  or  the  density  D  as  constant  :  but  if  the  differ- 
ent parts  of  the  line  be  supposed  unequally  dense,  the  quan- 
tity D  will  be  variable,  and  may  in  general  be  regarded  as  a 
function  of  x.  The  form  of  this  function  will  depend  on  the 
law  according  to  which  the  density  is  supposed  to  vary. 

589.  When  the  body  is  homogeneous,  it  is  frequently  con- 
venient to  regard  the  density  as  equal  to  unity  ;  and  the  factor 
D  is  then  replaced  by  1,  in  the  general  expression  for  the 
moment  of  inertia.  Having  determined  the  moment  of  inertia 
of  a  body  whose  density  is  equal  to  unity,  we  can  determine 
that  of  a  similar  body  whose  density  is  equal  to  D,  by  simply 
multiplying  the  former  moment  by  the  density  D.  In  the 
succeeding  examples,  we  shall  regard  the  density  as  equal  to 
unity. 

590.  As  a  second  example,  we  will  determine  the  moment 
of  inertia  of  the  area  of  a  circle  CBD  {Pig.  207),  ivith  refer- 
ence to  the  axis  AZ  passing  through  its  centre,  and  perfen- 
dicular  to  its  plane. 

Let  m  represent  a  point  in  the  plane  of  the  circle,  at  a  dis- 
tance mA=x  from  the  fixed  axis  :  the  areas  of  the  circles 
described  with  the  radii  x  and  x-\-dx  will  be  expressed 
respectively  by 

vx^,  and  7r(x-\-dxy  ; 
and  the  difference  between  these  areas,  by  neglecting  the  infi- 
nitely small  quantities  of  the  second  order,  will  be  25rx .  dx:. 
This  expression  will  represent  an  elementary  ring,  every 
point  of  which  will  be  at  the  distance  x  from  the  axis  :  hence, 
by  multiplying  this  element  by  x^,  we  shall  obtain  27rx^dxfoi 
the  differential  of  the  moment  of  inertia.    Taking  the  integral 


318  DYNAMICS. 

from  a?=:0  to  x='>'  we  shall  find  i^rr*  as  the  moment  of 
inertia  of  the  area  of  a  circle  whosk  radius  is  denoted  by  r. 

591.  Let  it  be  required  to  determine  the  iiioment  of  inertia 
of  a  sphere  with  reference  to  an  axis  passing  through  its 
centre.  If  the  sphere  be  cut  by  a  plane  EE'  perpendicular  to 
tlie  fixed  axis  AB  {Fig.  208),  the  section  will  be  a  circle 
whose  centre  will  be  fuund  at  the  point  D.  Denote  by  x  the 
absciss  AD  of  this  section,  and  by  y  the  ordinate  DE;  or  the 
radius  of  the  section.  The  moment  of  inertia  of  the  area  of 
this  circle  taken  with  reference  to  the  axis  AB,  will  be  expressed 
(Art.  590)  by 

and  if  this  expression  be  multiplied  by  dr=DD',  the  product, 

l^y'dx, 

will  express  the  moment  of  inertia  of  the  elementary  volume 
EE'F'F  bounded  by  parallel  planes  drawn  through  the  con- 
secutive points  D  and  D'.  The  integral  of  this  expression, 
being  taken  between  the  limits  :r=0  and  a:=AB=2r,  will 
give  the  moment  of  inertia  of  the  entire  sphere. 
But  by  the  property  of  the  circle,  we  have 

y 2  =2rx — x^  ; 
and  therefore, 

f^7ri/*dx=^^f(2rx—x''ydx 

=7rf(2r^x^  —2rx^-\-  ^x*)dx  ; 
or, 

f^^y'dx=^x^{"fr^-irx+-f\x^)-\-C. 
The  constant  C  will  be  equal  to  zero,  since  the  moment  is 
zero  when  x=0:  and  by  making  x=2r,  we  obtain  for  the 
moment  of  the  whole  sphere, 

_*_«■/•  ^. 

15'      • 

These  examples  are  sufficient  to  explain  the  manner  in 
which  the  determination  of  the  moment  of  inertia  is  reduced 
to  a  simple  problem  of  the  integral  calculus. 

592.  When  the  moment  of  inertia  of  any  body  with  refer- 
ence to  an  axis  passing  through  its  centre  of  gravity  has 
been  determined,  its  moment  with  respect  to  a  parallel  axis  is 
readily  found. 


MOMENT    OF    INERTIA.  319 

For  let  GF  and  CK  {Pig.  209)  represent  two  parallel  axes, 
the  first  of  which  passes  through  G,  the  centre  of  gravity  of 
a  body  :  let  the  origin  be  assumed  at  the  point  G,  the  line 
GF  being  the  axis  of  z.     Through  a  point  m,  assumed  arbi- 
trarily within  the  limits  of  the  body,  let  the  plane  mKF  be 
drawn,  parallel  to  the  plane  oi  x^y  \  this  plane  wilt  cut  the 
axes  GF  and  CK  at  two  points  F  and  K,  and  the  distances 
of  the  point  ni  from  these  axes  will  be  represented  respect- 
ively, by  the  right  lines  mK  and  mF,  which  we  shall  denote 
by  r  and  r'.     From  the  point  m  let  the  perpendicular  mE  be 
demitted  upon  the  plane  of  a;,  y  ;  the  triangles  ECG,  mKF  will 
be  equal  in  all  respects,  and  the  sides  of  the  former  may  there- 
fore be  substituted  for  those  of  the  latter.     Denote  by 
a,  and  j3,  the  co-ordinates  GD  and  DC  of  the  point  C, 
X  and  y,  the  co-ordinates  GP  and  PE  of  the  point  E, 
a,  the  distance  between  the  axes  : 
we  shall  have 

GC2=GD2.fDC=,     GE2=GP2-fPE2, 
or, 

«2  =^2  4-^2^        ^'2  ^j.2  J^y2 (332). 

Again,  the  right  line  CE  passing  through  points  whose 
co-ordinates  are  x  and  y,  «  and  /3  ;  the  value  of  CE— r  will 
result  from  the  equation 

or,  by  developing  the  terms  of  the  second  member, 
r2  =  r2  -f  y2  _2«.r— 2/3y+«2  4.^2  j 

and  reducing  by  means  of  equations  (33,";),  we  obtain 

multiplying  by  dM  and  integrating,  we  have 

fr''dM=fr'^d^l-2ccfx(M—2i2fydM.+a^fdM (333). 

The  expressions  fxdM.  and  fydM.  which  enter  into  this  equa- 
tion, are  equal  to  zero  ;  for,  let  x  and  y  represent  the  co- 
ordinates of  the  element  dM  of  the  mass  M  ;  the  moments  of 
this  element  with  reference  to  the  planes  ofx,  z  and  y,  z  will  be 
ydM  and  xdM.  :  hence,  the  co-ordinates  x,  and  y,  of  the  centre 
of  gravity  of  the  mass  M  will  be  determined  by  the  equations 
M.x=fxdM,     My=fydM.. 


320  DYNAMICS. 

But  in  the  present  instance,  the  centre  of  gravity  is  situated 
in  the  axis  of  z  ;  and  the  co-ordinates  x,  and  y,  are  therefore 
equal  to  zero  :  hence, 

/r^M=0,    fy(m.=0. 
Reducing  equation  (333)  by  means  of  these  values  and  sub- 
stituting M  for  its  equal  /c/M,  we  shall  obtain 

fr-dM^fr'^dM  +  Ma^ (334). 

The  expression  y?''2rfM  being  the  moment  of  inertia  with  re- 
ference to  the  axis  passing  through  the  centre  of  gravity,  we 
conclude  that  when  the  value  of  this  moment  has  been  found, 
that  of  the  moment  of  inertia /r^cZM,  taken  with  reference  to 
a  parallel  axis,  may  be  immediately  determined,  by  adding  to 
the  former  the  product  of  the  mass  of  the  body  by  the  square 
of  the  distance  between  the  two  axes. 

The  equation  (334)  may  be  written  under  the  form 


y-..<M=M(-qf+„=), 


and  this  expression  maybe  simplified  by  putting-^ — :^    =k^. 

Adopting  this  notation,  the  moment  of  inertia  taken  with 
reference  to  any  axis  will  be  expressed  by  the  formula 


Of  the  Motion  of  a  Body  about  a  Fixed  Axis  when  acted 
upon  by  Incessant  Forces. 

593.  Let  us  now  suppose  that  the  several  material  points 
of  a  system  which  is  retained  by  a  fixed  axis  kz  {Fig.  210), 
are  acted  upon  by  incessant  forces  :  each  particle  m  will 
describe  about  the  fixed  axis,  the  arc  of  a  circle  mno^  the 
plane  of  which  will  be  perpendicular  to  this  axis,  and  will 
intersect  it  at  a  point  C.  Let  <p  denote  the  incessant  force 
acting  upon  the  particle  w,  and  S"  the  angle  TmP  formed  by 
its  direction  with  the  tangent  to  the  circle  mno  at  the  point  m. 
The  force  4»  may  be  resolved  into  three  components  ;  one 
parallel  to  the  fixed  axis,  which  will  have  no  tendency  to  turn 
the  body  about  this  axis  ;  a  second  directed  along  the  radius 


VARIED   MOTION   ABOUT    AN   AXIS.  321 

mC,  which  will  be  destroyed  by  the  resistance  of  the  axis  ; 
and  a  third  coinciding  in  direction  with  the  element  of  the 
curve  described  by  the  particle  m  :  this  last  component  will 
be  expressed  by  (p  cos  ^,  and  will  be  the  only  portion  of  the 
force  <p  which  tends  to  turn  the  system  about  the  axis  Az. 

Let  ta  represent  the  angular  velocity  of  the  system  at  the 
expiration  of  the  time  /,  and  r  the  distance  Cm  of  the  particle 
m  from  the  axis  of  rotation  :  the  absolute  velocity  of  m,  at 
the  end  of  the  time  if,  will  be  expressed  by  ru  (Art.  579),  and 
in  the  succeeding  instant  dt,  this  velocity  will  be  increased  or 
diminished  by  the  action  of  the  incessant  force. 

If  the  particle  m  were  unconnected  with  the  other  parti- 
cles, the  force  4»  cos  ^  would  communicate  to  it  in  the  instant 
dt,  the  velocity  represented  by  <p  cos  ^.dt]  consequently,  the 
velocity  of  the  particle  m,  at  the  expiration  of  the  time  t  +  dt, 
would  be  expressed  by 

ra  +  4»  cos  S-.dt] 
but  this  particle  being  connected  with  the  other  parts  of  the 
system,  its  effective  velocity  at  the  end  of  the  time  t-{-dt 
will  actually  be  represented  by 

ro>-\-rda)  ; 
and  the  effective  quantity  of  motion  of  the  particle  m  will 
be  {7'ù)  +  rdw)m. 

The  same  remarks  being  applicable  to  the  other  particles 
which  compose  the  system,  it  is  necessary  that  the  quantities 
of  motion  impressed,  or 

2,[[ru-{-(p  cos  S'.dt)m] 
should,  by  the  principle  of  D'Alembert,  sustain  in  equilibrio 
the  effective  quantities  of  motion 

l,[(ru-\-rd<i))m\, 
the  latter  being  applied  in  directions  contrary  to  those  of  the 
motions  assumed. 

But,  in  order  that  an  equiUbriimi  may  subsist  between 
these  two  sets  of  forces,  it  is  necessary  that  the  sum  of  the 
moments  of  the  several  forces  taken  with  reference  to  the 
fixed  axis,  shall  be  equal  to  zero  :  and  since  these  forces  are 
exerted  in  the  directions  of  the  elements  of  the  circles  described 
by  the  material  points,  the  radii  of  these  circles  will  represent 

X 


322 


DYNAMICS. 


the  perpendiculars  demitted  upon  the  directions  of  the  several 
forces.     The  equation  of  the  moments  will  thus  become 

l.[{r''<v-\-r<p  cos  ^.  dt)m]—i:[{r'' u  +  r^ dc^)m]=0  ] 
or,  by  reduction, 

1(  r"  do).  m)=2(r<p  cos  S'.di.m) (335). 

The  quantities  dt  and  du  being  the  same  in  all  the  terms 
of  this  equation,  they  may  be  placed  without  the  sign  2  ;  and 
when  the  number  of  terms  is  regarded  as  infinite,  the  value 
of  each  being  infinitely  small,  the  character  2  may  be  re- 
placed by  the  integral  sign  f,  and  the  particle  m  by  dM,  the 
diiferential  of  the  entire  mass  :  thus  we  shall  have 

dtfr  .(pcos^.dM.  =do,/r^ dM  ; 
from  this  equation  we  deduce 

d<u_fr  .ç  cos  S'.dM.  cv^e^ 

dt  JFdM        ^       ''' 

To  complete  the  integrations  here  indicated,  it  is  necessary 
to  know  the  positions  of  the  elements  which  compose  the 
body,  and  the  directions  and  intensities  of  the  incessant  forces 
exerted  upon  each  particle.  These  particulars  will  be  exam- 
ined in  the  following  section. 

Of  the  Compound  Pendidum. 

594.  The  compound  pendulum,  represented  in  Fig.  211,  is 
composed  of  a  body,  or  a  system  of  material  points,  connected 
together  in  an  invariable  manner,  and  supported  by  a  hori- 
zontal axis  KL.  When  the  body  is  turned  around  this  axis, 
the  points  m,  m',  m",  &c.  describe  arcs  of  circles  7nn,  in'n', 
Qn"n",  &c.  ;  the  centres  of  these  circles  are  situated  in  the  axis 
KL,  and  their  planes  are  perpendicular  to  it. 

595,  The  motion  of  the  pendulum  being  referred  to  three 
rectangular  axes,  let  the  axis  of  z  be  supposed  to  coincide 
with  the  horizontal  line  Cz  {Fig.  212),  about  which  the  body 
turns,  and  the  axis  of  x  to  be  vertical  ;  the  plane  of  0,  y  will 
then  be  horizontal.  If  we  suppose  the  incessant  force  exerted 
upon  each  particle  to  be  that  of  gravity,  we  shall  have 

^=^'=^"=(fcc.=g-. 


COMPOUND    PENDULUM.  323 

The  direction  of  tlie  force  which  sohcits  a  particle  m,  being 
parallel  to  the  axis  of  .r,  the  intensity  of  this  force  may  be 
represented  by  a  portion  nig  of  a  vertical  line  ;  the  angle  <^ 
will  be  equal  to  ^mg  ;  and  if  the  perpendicular  mD  be  de- 
niitted  upon  the  axis  of  x,  the  angles  CmD  and  'Tmg  will  be 
equal  to  each  other,  being  each  the  complement  of  the  angle 
TmD:  hence,  C7nD=J';  and  consequently,  the  equation 

wiD=Cm  xcos  CmD 
will  become 

mD=Cm .  cos<^; 

or, 

y=r .  cos  <^. 

The  values  of  cos  ^  and  ç>  being  substituted  in  equation  (336), 
we  obtain 

dç^_/gpdM  . 
dt     fr^dM  ' 
or,  since  g  is  constant, 

dco^gfi/dM 
dt  .fr^dM^ 
The  expression  ydW  represents  the  moment  of  the  elementary 
mass  dM  taken  with  reference  to  the  plane  oî  x,  z\  if,  there- 
fore, we  denote  by  y,  the  distance  of  the  centre  of  gravity  of 
the  entire  mass  M  from  the  same  plane,  we  may  replace  fydM. 
by  My,,  and  the  preceding  equation  will  then  become 
d^_  gWy,  ^^ 

dt-J^l ^'^*^^^- 

and  since //'^6?M  expresses  the  moment  of  inertia  with  refer- 
ence to  the  axis  C2;,  this  moment  may  be  represented  (Art. 
592)  by  M(Â:2+a^).  Substituting  this  value  in  equation 
(337),  we  find 

^  =  ,  ^y'      (338). 

596.  It  has  been  shown  (Art.  592),  that  the  quantity  a  in 
the  expression  M(a-  +  k"^  )  represents  the  distance  CG  (Pig. 
209)  between  the  axis  CK  and  the  parallel  axis  GF  passing 
through  the-  centre  of  gravity.  But,  by  the  motion  of  the 
system,  the  centre  of  gravity  describes  a  circle  having  its 
radius  CG=a  {Fig.  213),  and  its  plane  arCL  perpendicular  to 


324 


DYNAMICS. 


the  axis  CK  ;  hence,  the  ordinate  DG  will  represent  the 
quantity  y,,  and  we  shall  have  from  the  property  of  the  circle, 

Again,  if  s  denote  the  arc  described  by  the  point  G,  the  velo- 

ds 
city  of  this  point  will  be  expressed  by  — -  :  but  this  velocity 

at 

will  also  be  expressed  by  a»  (Art.  579).     Hence,  we  shall 

have 

ds 

and  consequently, 

ds 
«= — —' 
adt 

The  values  of  a  and  y^  being  substituted  in  equation  (338), 
convert  it  into 

ade  kr-^w" 

597.  If  we  multiply  each  member  of  this  equation  by  2ac?5, 
the  first  member  will  become  an  exact  differential,  and  we 
shall  obtain  by  integration, 

%'  ""'  ""'  ^y^FS^^*^^^"^^'""^''^ ^^^^^' 

The  integral  of  the  second  member  can  only  be  obtained 
after  eliminating  one  of  the  two  variables  which  it  contains  : 
this  may  be  effected  by  means  of  the  equations 

ds=^{dx^-  -\-dy;-),     y=y/{2ax,—x;-)  ; 
and  by  proceeding  as  in  Art.  465,  we  find 
,  _       — adx, 

^(2ax,—x,^)  ' 
substituting  this  value  in  equation  (339),  we  have 

V2  =  —  /  ,  ^^  dx,  ; 

whence,  by  integration, 

^^^_2a2^        (340). 

To  determine  the  value  of  the  constant  C,  let  EB=6  repre» 


COMPOUND    PENDULUM.  325 

sent  the  value  of  .x\  at  the  instant  when  v~0  ;  the  supposition 
of  w=0  and  x=b  gives 


C  = 

2ar-gb  . 
k^-+a   ' 

and  the 

equation  (340) 

will  therefore  become 

v^,    or 

dp 

-'^"/)  ; 

whence, 

dt—       ■    7  ^ 

ds 

dt=- 


This  equation  can  be  readily  integrated  when  the  oscillations 
are  performed  through  very  small  arcs,  as  usually  happens  ; 

for,  by  replacing  ds  by  its  value  — -!_  obtained  on  the 

y/{2ax) 

supposition  that  x,  may  be  neglected  as  exceedingly  small  in 
comparison  with  2a,  in  the  expression 

»    — CiUiX I 

^~  y/(2ax,—x,^)' 
th^  equation  (341)  becomes 

jdx^ 

which  may  be  written  under  the  form 

di=-.^(^^l±^)  X     ,f  -    ,    , (342). 

'^    \     ag     J      ^[{b-x)x] 

598.  By  comparing  this  equation  with  the  equation  (228), 

it  will  appear  that  they  differ  only  by  the  constant  factor, 

/  ( k"^  _L^2\ 

which  in  the  former  is  \ \/  \ j ,  and  in  the  latter 

-\/ —     Hence,  the  integral  of  (342)  may  be  immediately 

obtained  from  that  of  (228),  the  constants  being  determined 
by  the  same  condition,  that  when  jf  =0,  x,=b.  Consequently, 
if  we  denote  by  I  the  length  of  a  simple  pendulum,  or  if  we 

replace  —  in  equation  (228)  by  -  -,  and  determine  I  by  the  con- 
dition  2§ 


326  DYNAMICS. 

g         «^ 
the  simple  pendulum  and  the  compound  pendulum  will  per- 
form their  oscillations  in   the  same  time.     The  preceding 
equation  gives 

a 
Thus,  by  means  of  this  formula  we  can  always  find  the  length 
of  the  simple  pendulum  which  will  perform  its  oscillations 
in  the  same  time  as  a  given  compound  pendulum. 

599.  If,  at  the  distance  I  from  the  axis  of  suspension  AB,  a 
line  EF  {Fig.  214)  be  drawn  parallel  to  the  axis  AB,  this 
parallel  will  enjoy  the  property,  that  all  points  contained  in  it 
will  perform  their  oscillations  in  the  same  time  as  though 
they  were  unconnected  with  the  other  points  of  the  body. 
When  the  line  EF  is  contained  in  the  plane  passing  through 
the  axis  of  suspension  AB  and  the  centre  of  gravity  of  the 
body,  this  line  is  called  the  aads  of  oscillation^  and  its  several 
points  are  called  centres  of  oscillation. 

600.  The  axes  of  suspejision  and  oscillation  are  recipro- 
cal ;  that  is  to  say,  if  we  take  the  axis  of  oscillation  EF 
{Fig.  214)  as  a  new  axis  of  suspension,  the  corresponding 
axis  of  oscillation  will  coincide  with  the  original  axis  of  sus- 
pension. 

To  demonstrate  this  property,  we  resume  the  expression  for 
CDj  the  distance  between  the  axes  of  suspension  and  oscilla- 
tion given  in  Art.  598, 

l^a^+k^ .343) 

a 

If  we  then  assume  the  line  EF  as  an  axis  of  suspension, 
and  represent  by  V  and  a'  the  corresponding  distances  of  the 
centres  of  oscillation  and  gravity  from  this  axis,  we  shall  have 
by  the  nature  of  the  centre  of  oscillation, 

n'-  -4-Z-2 

1'-=^    ^      (344). 

a' 

And  since  the  equation  (343)  indicates  that  the  distance  I 
exceeds  a,  it  follows  liiat  the  centre  of  gravity  will  be  situated 


COMPOUND    PENDULUM.  327 

between  the  axes  of  suspension  and  oscillation.    We  shall 
therefore  have  the  following  relation, 

a-{-a'=l, 
or, 

a'=l — a. 
By  means  of  this  value,  the  equation  (344)  becomes 

l,^(l-a)'-\-fc' .345)^ 

I— a 

Again,  from  equation  (343)  we  have 

7  ^' 

l^a= —  ; 

a 
and  the  value  of  I'  may  therefore  be  changed  into 


I' 


or,  by  reduction, 


l'='L  +  a=l 
a 


consequently,  when  the  line  EF  is  taken  as  the  axis  of  sus- 
pension, the  axis  of  oscillation  KH  is  situated  at  a  distance 
MX  from  the  line  EF,  precisely  equal  to  that  which  separates 
the  axes  AB  and  EF. 

601.  The  equation  (343)  gives 

a{l—a)=k^  ; 
and  by  replacing  I — a  by  its  value  a',  we  have 

aa!-=k^  : 
but  the  value  of /j^^  which  is  dependent  on  the  moment  of  in- 
ertia taken  with  reference  to  an  axis  passing  through  the  centre 
of  gravity,  and  parallel  to  the  axis  AB,  will  remain  constant 
so  long  as  the  direction  of  the  axis  remains  unchanged  :  hence 
it  appears  that  if  the  body  be  caused  to  oscillate  about  any 
axis  parallel  to  AB,  and  at  a  distance  from  the  centre  of 
gravity  represented  by  a,  the  corresponding  axis  of  oscillation 
will  be  found  at  a  distance  a'  from  the  centre  of  gravity  ;  thus 
the  value  of  a-\-a\  or  the  length  of  the  equivalent  simple 
pendulum,  will  be  the  same  as  when  the  oscillations  were  per- 


328  DYNAMICS. 

formed  about  the  axis  AB.  A  similar  remark  is  applicable  to 
all  those  axes  parallel  to  AB  which  are  situated  at  a  distance 
a'  from  the  centre  of  gravity.  If,  therefore,  the  body  be  sus- 
pended successively  from  any  number  of  axes  parallel  to  AB, 
and  at  a  distance  from  the  centre  of  gravity  equal  to  a  or  a', 
the  times  of  oscillation  about  such  axes  will  be  equal  to  each 
other. 

These  parallel  axes  of  suspension  about  which  the  oscilla- 
tions are  performed  in  equal  times,  will  evidently  be  found 
in  the  surfaces  of  two  cylinders  having  a  common  axis  pass- 
ing through  the  centre  of  gravity. 

602.  The  expression  for  the  distance  I  between  the  axes  of 
suspension  and  oscillation  may  be  put  under  the  form 

Wi  Ma     ' 

and  since  this  value  is  precisely  equal  to  that  which  was 
obtained  for  the  distance  of  the  centre  of  percussion  from  the 
axis  of  rotation  (Art.  585),  it  appears  that  the  centre  of  per- 
cussion, when  it  exists,  will  be  found  upon  the  axis  of 
oscillation. 

Of  the  Motions  of  a  Body  in  Space  when  acted  upon  by 
Impidsive  Forces. 

603.  In  the  preceding  sections,  the  circumstances  of 
motion  of  a  body  retained  by  a  fixed  axis  have  been  alone 
discussed  ;  it  now  becomes  necessary  to  consider  the  motions 
of  a  body  in  space  when  unconnected  with  fixed  objects. 

IjCt  m,  ni',  m",  (fcc.  represent  material  points  composing  a 
system  whose  several  particles  are  unconnected,  and  let  v,  v', 
v",  &c.  represent  the  velocities  respectively  impressed  upon 
these  particles  in  directions  parallel  to  each  other  :  it  is 
required  to  determine  the  motion  of  the  common  centre  of 
gravity  of  the  system. 

If  a  plane  be  passed  through  the  primitive  position  of  the 
centre  of  gravity  parallel  to  the  common  direction  in  which 
the  impulses  are  applied,  the  sum  of  the  moments  of  the 
particles  m,  m',  rii'%  (fcc,  taken  w\û\  reference  to  this  plane, 
Mall  be  equal  to  zero  at  the  commencement  of  tlie  motion  ; 


PERCUSSION.  329 

and  it  is  likewise  evident  that  this  sum  will  remain  equal  to 
zero  during  the  motion,  since  the  distances  of  the  bodies  from 
the  assumed  plane  remain  invariable.  Hence,  the  motion  of 
the  centre  of  gravity  will  be  confined  to  this  plane  ;  and  since 
the  same  may  be  said  of  any  other  plane  drawn  through  the 
primitive  position  of  the  centre  of  gravity  and  parallel  to  the 
direction  of  the  motions,  it  follows  that  the  centre  of  gravity 
will  continue  in  each  of  these  planes,  or  in  their  line  of  inter- 
section ;  and  we  therefore  conclude  that  the  motion  of  the 
centre  of  gravity  of  such  a  system  is  rectilinear ,  and  parallel 
to  the  direction  of  the  m,otions  of  its  several  parts. 

Let  a  plane  be  drawn  perpendicular  to  the  direction  in 
which  the  bodies  move,  and  represent  the  distances  of  the 
several  bodies  from  this  plane  at  the  commencement  of  the 
motion,  by  S,  S',  S",  &c.  :  their  distances,  at  the  expiration  of 
the  time  t,  will  be  expressed  by 

S+î)^,     S'-f  v'/,     S"+D'7,  (fee. 
If  a  and  x,  represent  the  distances  of  the  centre  of  gravity 
of  the  system  from  the  perpendicular  plane,  at  the  commence- 
ment of  the  motion,  and  at  the  end  of  the  time  t,  we  shall 
have,  by  the  property  of  the  centre  of  gravity, 

wS -{- m'S' -j- m"S" + «fee.  =  (m  4- ^W'' •  I- W  4- <fec.  )a, 
m{^^-vt)-\-m'{^'  -{-v't)-[-m"{^"  ■\-v"t)-\-&^c,= 
{m-\-m' -\-'in" -\-ÔLc)x ,  ; 
and  by  subtraction,  we  obtain 

{m-\-m'  -\-m"  -\-éLC.)[x, — a)={niv-\-m'v'-'rm,"v"-\-&Lc)t: 
hence,  it  appears  that  the  space  passed  over  by  the  centre  of 
gravity  is  proportional  to  the  time,  or  the  motion  of  the  centre 
of  gravity  is  uniform. 

It  is  to  be  understood  that  those  velocities  are  regarded  as 
negative,  whose  directions  are  opposite  to  such  as  we  con- 
sider positive. 

604.  The  preceding  equation  may  be  written  under  the 
form 


^x—a 
X, — a 


{m-{-m!  -{-mf'-^-oi^c)— =wu + wV-f- wV4-<fec.  ; 


the  expression  — represents  the  velocity  of  the  centre  of 


330 


DYNAMICS. 


gravity,  and  is  independent  of  the  positions  of  the  particles 
m,  m',  m",  <fcc.,  to  which  the  quantities  of  motion  mv,  Tti'v', 
tn"v"^  &c.  are  respectively  apphed  :  it  follows,  therefore,  that 
if  we  suppose  a  mass  M  equal  to  the  sum  of  the  masses  m, 
m\  7Ji'\  (fee.  to  be  concentrated  at  the  centre  of  gravity,  the 
quantity  of  motion  of  this  mass  will  be  equal  to  the  sum  of 
the  quantities  of  motion  in  the  entire  system. 

We  also  conclude,  that  the  centre  of  gravity  will  have  the 
name  Tnotion  as  though  the  several  masses  m,  m\  m",  Sf'c. 
were  concentrated  in  this  jjoint,  and  the  several  forces  applied 
immediately  to  it  in  directions  parallel  to  those  along  which 
they  were  originally  applied. 

605.  When  the  forces  applied  to  the  different  particles  are 
not  parallel,  ;hey  may  be  resolved  into  components  parallel  to 
three  rectangular  axes,  and  since  the  effects  produced  by  each 
system  of  parallel  components  will  be  independent  of  the 
other  two  systems,  it  may  in  like  manner  be  shown  that  the 
motion  of  the  centre  of  gravity  parallel  to  each  of  the  axes 
will  be  uniform,  and  equal  to  that  which  would  be  produced 
by  concentrating  the  masses  at  the  centre  of  gravity,  and 
applying  the  several  forces  directly  to  that  point. 

606.  Let  the  several  masses  be  now  supposed  connected  in 
an  invariable  manner,  the  same  property  will  be  equally  true. 
For,  let  mv,  mv',  m"v",  (fee.  represent  the  quantities  of  motion 
impressed  upon  the  particles  m,  m',  m!',  (fee,  and  let  each  of 
these  quantities  of  motion  be  resolved  into  components  mu 
and  wU,  (fee,  the  first  of  which  shall  be  the  effective  quantity 
of  motion  retained  by  the  particle,  the  second  being  'Ipstroyed 
by  the  mutual  connexion  of  the  parts  of  the  systein  :  then, 
since  the  quantities  of  motion  mxi,  m!u',  m"u",  (fee,  commu- 
nicated to  the  masses  m,  m',  m!',  (fee,  produce  their  full  effects, 
these  masses  will  move  under  their  influence,  in  the  same 
manner,  whether  we  regard  them  as  free  or  connected. 

Hence,  it  appears  that  the  centre  of  gravity  of  the  system 
will  move  in  the  same  manner  as  though  the  quantities  of 
motion  mn,  mJii!,  in"u",  (fee.  were  applied  directly  to  it.  The 
quantities  of  motion  ii\\},  m'\}',  m"U",  (fee.  being  such  as  to 
destroy  each  other  when  applied  to  the  different  points  m,  vi'^ 


PERCUSSION.  331 

tn"^  (fcc,  they  must  (Arts.  54  and  130)  destroy  each  other 
when  appHed  to  the  centre  of  gravity. 

But  the  two  systems  mi/,  m'u'j  m"u",  (fcc,  WiU,  wt'U',  m"U", 
«fee,  may  be  replaced  by  the  original  system  mv,  m'v',  m,"v'\ 
&.C.,  and  we  therefore  conclude  that  the  centre  of  gravity  will 
have  the  same  motion  as  though  the  several  masses  had  been 
concentrated  at  that  point,  and  the  original  quantities  of 
motion  niv,  m'v',  m"v",  ^-c.  impressed  immediately  upon  it. 

607.  If  an  impulse  P  be  communicated  to  any  point  of  a 
body  in  a  direction  not  passing  through  the  centre  of  gravity, 
this  centre  will  assume  a  motion  precisely  equal  to  that  which 
would  have  been  produced  by  the  direct  application  of  the 
force  to  it.  But  a  motion  of  rotation  will  also  be  commu- 
nicated to  the  body  ;  for,  if  an  equal  force  Q,  [Fig.  215)  be 
applied  to  the  centre  of  gravity  in  a  parallel  and  opposite 
direction,  the  joint  action  of  the  two  forces  P  and  Q,  will 
maintain  the  centre  of  gravity  at  rest.  From  the  centre  of 
gravity  G  demit  the  perpendicular  GA  upon  the  direction  of 
the  force  P,  and  lay  off  on  the  opposite  side  of  the  point  G  a 
distance  GB=AG.  liCt  the  force  d  be  then  resolved  into 
two  components,  each  equal  to  iQ,  or  |P,  applied  at  the 
points  A  and  B.  The  forces  P  and  \Q,  applied  at  the  point 
A,  and  acting  in  contrary  directions,  will  have  a  resultant 
equal  to  ^P  ;  thus  the  body  will  be  acted  on  by  two  forces 
each  equal  to  |P,  acting  at  the  distance  AG=BG  from  the 
centre  of  gravity,  and  tending  to  turn  the  body  about  that 
point.  And  since  the  point  G  may  be  regarded  as  fixed,  the 
two  forces  will  have  the  same  effect  to  turn  the  body  about 
that  point  as  the  single  force  P  acting  at  A.  The  effect  of 
the  force  d  will  be  simply  to  destroy  the  motion  of  transla- 
tion, without  affecting  the  motion  of  rotation. 

Hence  we  conclude^  that  when  a  body  receives^  an  impulse 
in  a  direction  which  does  not  pass  through  the  centre  of  gra- 
vity, that  centime  icill  assume  a  motion  of  translation  as  though 
the  impidse  were  applied  i^nmediately  to  it  ;  and  the  body 
will  likeioise  have  a  ynotion  of  rotation  about  the  centre  of 
gravity,  as  though  that  point  were  immoveable. 

608.  The  circumstances  of  motion  of  a  body  which  is 
divided  symmetrically  by  a  plane  passing  through  the  direc- 


332  DYNAMICS. 

tion  of  the  impulse  can  now  be  readily  determined.  For,  the 
motion  of  translation  of  the  centre  of  gravity  will  be  similar 
to  that  of  a  material  point  to  which  an  impulse  is  applied  ; 
and  the  motion  of  rotation  being  precisely  the  same  as  that 
which  would  take  place  if  a  fixed  axis  passed  through  the 
centre  of  gravity,  perpendicular  to  the  dividing  plane,  it  will 
merely  be  necessary  to  apply  the  results  obtained  in  Arts.  581 
and  582. 

Let  Mv  represent  the  quantity  of  motion  impressed  upon  a 
body  whose  mass  is  represented  by  M  (Fig.  216),  a.ndp  the 
perpendicular  distance  from  the  centre  of  gravity  G  to  the 
line  of  direction  of  the  impulse.  The  centre  of  gravity  will 
assume  a  uniform  motion  with  the  velocity  v,  in  a  direction 
parallel  to  that  of  the  impulsive  force.  The  angular  velocity 
will  result  immediately  from  equation  (331),  and  will  be 
expressed  by 

Mvp vp 

609.  The  absolute  velocity  of  each  point  of  the  body  will 
be  compounded  of  the  two  velocities  of  translation  and  rota- 
tion. Thus,  the  point  O,  for  example,  to  which  the  force  is 
applied,  has  two  velocities  ;  a  velocity  of  translation  Oi  equal 
to  that  of  the  centre  of  gravity,  and  a  velocity  of  rotation  ih 
about  that  point  ;  so  that  if  we  assume  any  point  on  the  line 
OGC,  at  a  distance  a  from  the  centre  of  gravity,  its  velocity 
will  be  expressed  by  v±a6>:  the  superior  sign  applies  to 
those  points  which  are  situated  upon  the  same  side  of  the 
centre  of  gravity  as  the  point  O  ;  and  the  inferior  sign  to 
points  situated  on  the  opposite  side. 

610.  If  we  consider  the  motion  of  the  point  O  for  an  ex- 
ceedingly short  interval  of  time,  the  path  Oih  described  by 
this  point,  whilst  the  centre  of  gravity  describes  the  line  GG', 
may  be  regarded  as  a  right  line  :  thus,  the  line  OGC  will 
assume  the  position  AG'C,  the  point  C  remaining  at  rest 
during  this  interval.  This  point  is  called  the  centre  of  spon- 
taneous Isolation  :  its  position  may  be  determined  by  the  con- 
dition that  its  velocity  of  rotation  shall  be  equal  to  that  of 
translation  :  indeed,  whilst  the  point  C  would  be  carried  for- 
ward over  the  line  CC  by  the  motion  of  translation,  it  would 


FREE    MOTION    OF    A    SYSTEM,  333 

be  moved  backward  through  the  same  distance  by  the  motion 
of  rotation  :  this  condition  will  give  the  absolute  velocity  of 
the  point  C 

V — aa>=0  ; 
whence, 

and  we  therefore  have 

OC=OG  +  GC=»+a-=»  +  — ; 

P 
from  which  we  conclude,  that  the  centre  of  spontaneous  rota- 
tion will  coincide  with  the  centre  of  percussion,  if  the  axis  of 
rotation  he  supposed  to  pass  through  the  point  O. 

611.  When  the  plane  passing  through  the  direction  of  the 
impulse  and  the  centre  of  gravity  divides  the  body  into  two 
portions  which  are  not  symmetrically  situated  with  respect 
to  this  plane,  it  will  usually  occur  that  the  axis  about  which 
the  body  revolves  will  not  retain  an  invariable  position. 
For,  the  rotatory  motion  of  the  body  will  develop  in  each 
particle  a  centrifugal  force,  producing  a  pressure  upon  the 
axis  ;  and  unless  these  pressures  are  such  as  to  destroy  each 
other,  the  direction  of  the  axis  will  necessarily  be  changed. 

Of  the  Motions  of  a  System  in  Space  when  acted  upon  by 
Incessant  Forces. 

G12,  We  will  next  investigate  the  circumstances  of  motion 
in  a  system  whose  different  particles  are  acted  upon  by  inces- 
sant forces.  Let  the  force  acting  on  a  particle  m  be  resolved 
into  three  components  X,  Y,  Z,  respectively  parallel  to  three 
rectangular  axes  ;  that  acting  on  m'  into  the  three  X',  Y',  Z', 
(fee.  Let  a,  6,  and  c  represent  the  variable  co-ordinates  of  the 
centre  of  gravity  referred  to  the  fixed  axes,  and  let  three  axes 
be  drawn  through  the  centre  of  gravity,  parallel  to  the  fixed 
axes,  and  moveable  with  the  system  in  space.  Then,  if  x,  y, 
z,  x',  y\  z\  &c.  denote  the  co-ordinates  of  the  points  m,  m\ 
m",  &c.  referred  to  the  moveable  axes;  a+a:,  6-fy,  c  +  5?, 
a-\-x',  h-\-y\  c-\-z',  &c.  will  express  the  co-ordinates  of  the 
same  points  when  referred  to  the  fixed  axes. 


334  DYNAMICS. 

613.  The  velocity  of  the  particle  m  in  the  direction  of  the 

axis  of  X,  at  the  expiration  of  the  time  t^  will  be  expressed  by 

dla-\-x)     da-\-dx 

v=—^ = : 

dt  dt      ' 

and  in  the  succeeding  instant  dt,  this  velocity  would  receive 

the  increment  Xdt,  by  the  action  of  the  incessant  force  X,  if 

the  praticle  m  were  entirely  free  ;   but  in  consequence  of  the 

connexion  existing  between  the  diiferent  parts  of  the  system, 

the  effective  velocity  communicated  to  the  particle  m  in  the 

time  dt,  will  be  expressed  by 

,        ,  da-\-dx 

dv—d  ; . 

dt 

and  the  velocity  destroyed  in  the  particle  m,  by  the  connexion 

of  the  parts  of  the  system,  will  therefore  be 

Xdt-d  ^^. 
dt 

The  same  remarks  being  applicable  to  the  velocities  parallel 

to  the  axes  of  y  and  z,  we  shall  have  for  the  quantities  of 

motion  destroyed  in  the  particle  m,  parallel  to  the  three  axes, 


m\ 


m 


Similar  expressions  may  in  like  manner  be  obtained  for  the 
quantities  of  motion  lost  by  the  other  particles  ;  and  we  shall 
therefore  obtain,  for  the  sum  of  the  quantities  of  motion  lost 
parallel  to  the  axis  of  x, 

2[m(xdt-d  ^^'^)] (346); 

or,  by  completing  the  differentiation  indicated,  regarding  dt 
as  constant,  we  have 

In  like  manner,  the  sums  of  the  quantities  of  motion  lost  in 
directions  parallel  to  the  axes  of  y  and  z,  will  be  expressed  by 

.[„(y..-'^/L^)] (34r), 


FREE    MOTION   OF   A    SYSTEM.  335 

4.(z..-*£±*f)] (348) 

The  quantities  of  motion  (346),  (347),  (348),  or  the  forces 
capable  of  producing  them,  being  such  as  to  destroy  each 
otlier,  they  must  satisfy  the  general  equations  of  equilibrium 
(66)  and  (67),  which  appertain  to  a  system  of  forces  having 
various  directions  and  applied  to  different  points  of  a  body. 

The  equations  (66)  indicate  that  the  sum  of  the  components 
parallel  to  each  of  the  axes  will  be  equal  to  zero  ;  we  shall 
therefore  have  for  those  components  parallel  to  the  axis  of  a: 


;Kx<._^ii^^)]=0; 


(349  a)  : 
(349  b). 


or,  by  multiplying  by  dt,  and  changing  the  form  of  the  expres- 
sion, we  have 

Q={mX-{-m'X'-\-m"X"-\-iSLC.)di^  —d^  a(m+m'+m"+&c.) 

—{md^x+m'd'x'+m"d^x"-\-&c.) (349). 

But,  by  the  nature  of  the  centre  of  gravity, 
mx+m'x'-\-m"x"-{-ôcc.=0  ) 
my-\-m'i/'-\-m"y"-\-ôùC.=0  )  '    ' 
and  by  differentiating  twice,  we  find 

md'x+m'd^x'-{-m"d^x"-\-ôùC.=0 
md'i/+m'd''y'-\-m"d'  y"+&c. = 0 

The  first  of  these  values  being  substituted  in  (349),  and  the 
mass  of  the  system  being  denoted  by  M,  there  will  result 

Md^a=(mX+m'X'-\-m"X"+&c.)dt', 
or, 

M^=2(mX): 
dt^        ^        ' 

the  same  being  true  with  respect  to  the  components  parallel 

to  the  axes  of  y  and  z^  we  shall  obtain,  for  the  three  first 

equations  expressing  the  circumstances  of  motion  of  the 

system, 

M^=.(»X)  1 


i 


M^-2(mY)  \ (350). 

MÇ^=2(mZ) 


336  DYNAMICS. 

These  equations  serve  to  determine  the  motion  of  the  centre 
of  gravity  of  the  mass  M;  for  when  integrated,  they  will  ex- 
press the  velocities  -r  >  ;r  j  ;t-  ^^f  the  centre  of  gravity,  par- 
allel to  the  three  axes. 

614.  The  equations  (350)  make  known  a  remarkable 
property  of  the  centre  of  gravity.  For,  let  the  particles  m, 
m\  m'\  (fee  be  supposed  concentrated  at  their  common  centre 
of  gravity,  and  let  the  forces  wiX,  ?/iY,  mZ,  w'X',  m'Y',  m'7i\ 
&c.  be  applied  directly  to  that  point,  parallel  to  their  original 
directions.  These  forces  may  be  reduced  to  three,  MX,,  MY,, 
MZ„  the  values  of  which  will  result  from  the  equations 

MX=2(mX),     MY=2(mY),     MZ=2(mZ.) 
Eliminating  the  second  members  of  these  equations  by  means 
of  equations  (350),  we  have 

^=^»  1^'=^»  '^-^ (^^'>- 

But  when  the  forces  MX,,  MY,,  MZ,  are  applied  to  the  centre 
of  gravity  regarded  as  a  material  point  whose  mass  is  M,  the 
circumstances  of  its  motion  are  expressed  by  the  equations 
(180),  which  are  precisely  similar  to  the  equations  (351)  ; 
hence,  we  conclude  that  the  centre  of  gravity  of  the  system 
has  the  same  motion  as  though  the  forces  were  applied  directly 
to  that  point. 

615.  To  determine  the  circumstances  of  motion  of  the 
several  particles  m,  ra\  m",  cfec.  with  respect  to  the  centre  of 
gravity,  we  resume  the  equations  (67),  which  express  the 
conditions  that  the  forces  have  no  tendency  to  turn  the  sys- 
tem about  either  axis  :  that  this  may  be  the  case,  it  is  neces- 
sary that  the  sum  of  the  differences  of  the  moments  of  the 
components  parallel  to  any  two  of  the  axes,  as  x  and  y,  taken 
with  reference  to  the  corresponding  planes  of  y,  z  and  a-,  z, 
should  be  equal  to  zero.  But  if  we  consider  the  particle  m, 
the  distance  of  the  component  X,  which  acts  upon  it,  from 
the  plane  of  a-,  z  will  be  equal  to  y-f6,  the  co-ordinate  of  the 
point  m,  parallel  to  the  axis  of  y  :  in  like  manner,  the  distance 
of  the  force  Y  from  the  plane  of  y,  z  will  be  expressed  by 
x+a:  we  shall  therefore  have,  for  the  difference  of  the 
moments, 


TU 


FREE    MOTION   OF   A   SYSTEM.  337 

The  same  remarks  being  applicable  to  the  particles  m\  m", 
&c.,  we  shall  obtain  a  similar  expression  for  each.  By- 
placing  the  sum  of  these  expressions  equal  to  zero,  as  in 
equation  (67),  performing  the  multiplications,  and  reducing 
by  means  of  equations  (349  a)  and  (349  6),  we  shall  obtain 

62(mX)  -M6|^+2(myX)  -2  (^y^) 

— a2(mY)+Ma^-2(ma:Y)+2^w^:^•^^  =0. 

This  equation  admits  of  simplification  ;  for,  if  we  multiply 
the  first  of  equations  (350)  by  6,  and  the  second  by  a,  and 
take  their  difference,  we  shall  have 

62(mX)-a2(mY)— Mô^  +  M«— =0. 

This  relation  reduces  the  previous  equation  to 

2(myX)-2(m;2;Y)  -2  (^y^)  +2  {jnx^^  =0  ; 

whence, 

1  (m  î^!ty^)  =x[«(Yx-Xy)  A]. 

The  integral  of  the  first  member,  taken  with  reference  to  the 


time  L  is 


;(m-!^^^^): 


and  by  adopting  the  same  process  with  reference  to  the  other 
two  axes,  putting,  for  brevity, 

l.[mf(Yx-^Xy)dt]='L, 
j\mf{Zx—Xz)dt\='^, 
^mf{Zy-Yz)dt\=^, 
we  shall  obtain  the  three  equations  of  motion 

^/   ri^-^\     ^    |. (351a). 


Y  29 


J 


338  DYNAMICS. 

The  equations  (351  a)  are  independent  of  the  co-ordinates  of 
the  centre  of  gravity,  and  would  undergo  no  change  if  forces 
were  appUed  at  that  point  sufficient  to  destroy  its  motion  of 
translation,  since  such  forces  would  not  enter  into  the  ex- 
pressions L,  M,  and  N  ;  thus,  the  motion  of  rotation  about 
the  centre  of  gravity,  determined  by  these  equations,  is  pre- 
cisely similar  to  that  which  would  take  place  if  the  centre  of 
gravity  were  immoveable. 

Hence  we  conclude,  that  whe?i  arvy  body  is  acted  upon  hy 
incessant  forces  applied  to  its  several  particles,  the  body  will 
receive  two  motioiis  :  one  of  translation,  in  virtue  of  which  its 
centre  of  gravity  2vill  be  transported  in  space  as  though  the 
forces  were  apjjlied  directly  to  that  point  ;  and  a  seco?id,  of 
rotation  about  the  centre  of  gravity,  as  though  that  point  were 
absolutely  at  rest. 


General  Equations  of  the  Motions  of  a  System  of  Bodies. 

616.  Let  Z,  /',  l",  &.C.  represent  the  velocities  lost  or  gained 
by  the  several  material  points  which  compose  a  system,  in 
consequence  of  the  mutual  connexions  of  its  parts  ;  the  cor- 
responding quantities  of  motion  lost  or  gained  will  be  ml, 
m'l',  m'l",  «fcc,  and,  by  the  principle  of  D'Alembert,  these 
quantities  of  motion,  when  impressed  upon  the  particles 
m,  m',  m",  &c.  are  such  as  will  produce  an  equilibrium  :  hence, 
they  must  fulfil  the  conditions  of  equilibrium  expressed  in 
equations  (66)  and  (67). 

The  components  of  these  quantities  of  motion,  or  the 
forces  capable  of  producing  them,  estimated  in  the  directions 
of  three  rectangular  axes,  will  be 

ml  cos»,      7w/cos/3,'        ml  cos  y     components   of  ml. 

m'l'  cos  »,     m'l'  cos  /3',      m'l'  cosy' components  of  m'l'. 

m"l"  cos  »",  m"r  cos  /3",    m"l"  cos  y" components  oîm"l". 

(fee.  (fee.  &c.  (fee. 

We  shall  therefore  have  for  the  equations  of  equilibrium, 
2(mZ.  cos  «)=0  i 

2(mZ.cos(s)=:0  V (352). 

1{m.l .  cosy)=0  J 


FREE    MOTION   OF   A    SYSTEM.  339 

^[ml{x  COS  /3  — y  cos  a)]  =0  i 

^ml(z  COS  »—x  COS  y)]  =  0  > (353). 

2[m%  cos  y — 2;  cos /3)]  =0  ^ 

617.  If  the  system  is  retained  by  a  fixed  point,  the  three 
equations  (352)  cease  to  be  necessary  ;  the  equations  (353) 
being  alone  sufficient,  provided  the  origin  be  placed  at  the 
fixed  point. 

618.  When  there  are  two  fixed  points  within  the  system, 
we  connect  them  by  a  right  line,  and  assume  this  line  as  one 
of  the  co-ordinate  axes,  z  for  example  ;  the  first  of  equations 
(353)  will  then  be  sufficient  to  ensure  the  equilibrium  (Arts. 
132  and  133). 

619.  The  velocities  lost  or  gained  are  here  indicated  by  the 
letters  /,  l',  I",  &.c.  ;  but  to  express  these  quantities  in  functions 
of  the  incessant  forces  which  solicit  the  several  material 
points,  we  shall  first  consider  the  particle  m,  and  suppose  that 
the  forces  acting  upon  this  point  have  been  reduced  to  three, 
X,  Y,  and  Z,  respectively  parallel  to  the  co-ordinate  axes. 
The  velocity  of  the  particle  m,  parallel  to  the  axis  of  x,  at  the 

expiration  of  the  time  t,  will  be  expressed  by  -j-  (Art.  430)  ; 

and  at  the  end  of  the  time  t-{-di,  this  velocity  will  become 

-^J^d — ;    this  will    be    the  expression  for    the  effective 
dt       dt'  ^ 

velocity  of  the  particle  m. 

But  if  the  particle  m  were  perfectly  free,  the  incessant  force 
X  would  communicate  to  it  in  the  time  dt,  a  velocity  repre- 
sented by  ILdt  (Art.  391),  and  the  velocity  of  m  at  the  expira- 

tion  of  the  time  ^4-<^^j  would  be  expressed  by  — +Xd/; 

(XZ 

hence,  the  velocity  lost  or  gained  by  the  particle  m  will  be 
equal  to 

dx  ,  ■^j.     (dx     jdx\ 

and  by  reduction,  we  shall  find  that  Xc?/— rf^— will  express 

the  velocity  lost  or  gained  by  the  particle  m,  in  the  direction 

of  the  axis  of  x.     This  velocity  being  multiplied  by  the  mass 

m,  gives 

Y2 


840 


m 


DYNAMICS. 


(x..-.^), 


for  the  quantity  of  motion  lost  or  gained  by  m,  in  the  direc- 
tion of  the  axis  of  x  :  we  shall  therefore  have 


ml .  cos  ct=m  {  Xdt r—  Ï 

V  dt  / 


(354). 


In  like  manner,  by  considering  the  velocities  lost  by  m,  in 
directions  parallel  to  the  axis  of  y  and  z^  we  shall  find 

ml.  cos  ^=m  I Y  dt — -^j (355). 

7nl.  cosy=m(Zdt—-—\ (356). 

Similar  expressions  may  be  obtained  for  the  quantities  of 
motion  lost  or  gained  by  the  particles  m',  m",  «fcc.  ;  and  by 
including  their  sums  under  the  sign  2,  the  equations  (352) 
and  (353)  may  be  reduced  to 

.(.|f)=.(.X)' 

2(m^)  =2(mY)  [ (357). 

2(.|£)=.(.z)_ 


de  ^  ^ 

iK5^!^:=^^^=2[m(X;2-Z:r)]  . 
dt^ 

^^^(y^y-^^'^J^2[m(Zy-Yz)] 

CLZ 


(358). 


Such  are  the  most  general  forms  of  the  equations  expressing 
the  circumstances  of  motion  of  a  system. 

620.  The  expressions  Ya:— Xy,  X2; — Zx,  7,y—Yz,  &c. 
become  equal  to  zero  under  the  following  circumstances  :  1°. 
when  the  incessant  forces  acting  on  the  particles  m,  m',  m", 
(fee.  are  equal  to  zero  ;  2°.  when  all  the  forces  are  directed 
towards  the  origin  of  co-ordinates  :  3°.  when  the  forces  are 
such  as  arise  from  the  mutual  attractions  of  the  différent  parts 
of  the  system. 


FREE    MOTION  OP    A   SYSTEM.  341 

In  the  first  case,  the  incessant  forces  being  equal  to  zero, 
their  components  must  likewise  be  equal  to  zero  ;  and  hence 

X=0,     Y=0,     Z=0,     X'=0,  &c.: 
the  second  members  of  equations  (358)  will  therefore  dis- 
appear. 

621.  The  second  members  will  likewise  disappear,  when 
the  forces  are  directed  towards  the  origin  of  co-ordinates. 
For,  it  has  been  shown  (Art.  436),  that  when  the  fixed  point 
towards  which  the  forces  are  directed  does  not  coincide  with 
the  origin  of  co-ordinates,  if  we  represent  by  a,  6,  and  c  the 
co-ordinates  of  this  point,  and  by  jo,  p',  /?",  &c.  the  distances 
of  the  several  particles  from  the  fixed  point,  the  components 
of  the  forces  P,  P',  P",  (fcc,  in  the  directions  of  the  co-ordi- 
nate axes,  will  be  expressed  by 

p  p  p 

p  p  p 

P^     P'^-=^,     P"f^,&c.; 
p  p  p 

but,  by  hypothesis,  the  origin  coincides  with  the  fixed  point 

towards  which  the  forces  are  directed,  and  we  therefore  have 

a=C,     h=%     c=0: 

hence,  the  preceding  expressions  are  reduced  to 


Vx 

V'x' 

V"x" 
P"'- 

,  &c.. 

Py 

py 

P"y" 

&c., 

Vz 

Vz' 

F"z" 

&p 

v' 

p' 

P"' 

And  by  substituting  these  values  of  the  components  for  X, 
X',  X",  Y,  Y',  Y",  Z,  Z',  Z",  &c.  in  the  expressions 

Yx—Xy,  Xz-Zx,   Zy-Yz,  YV-Xy,  &c (359), 

we  shall  find  each  of  these  expressions  equal  to  zero.  Con- 
sequently, when  the  incessant  forces  which  act  upon  the 
several  particles  are  constantly  directed  towards  the  origin, 


342  DYNAMICS. 

the  expressions  (359)  become  equal  to  zero,  and  the  second 
members  of  equations  (358)  will  therefore  disappear. 

622.  The  same  consequences  may  be  deduced  when  the 
material  particles  are  subjected  only  to  their  mutual  attrac- 
tions. For,  by  putting  the  second  members  of  the  equations 
(358)  under  the  following  forms  : 

m(Y.-r— X2/)  +  m'(YV-Xy)+(fcc.  ^ 

m(X2;-Za:)+m'(X';s'-ZV)+&c.  > (360), 

m(Zy  —  Yz)  +  m'{Z'y'-Y'z')-[-&.c.  ) 
and  considering  the  material  points  two  by  two,  it  is  evident 
that  the  moving  force  exerted  by  the  point  m  upon  m'  is  equal 
to  that  exerted  by  m'  upon  m.  Hence,  if  X,  Y,  Z,  X',  Y',  Z', 
&c.  represent  the  components  of  the  incessant  forces  P,  P, 
P",  (fee,  we  shall  have 

w'X'= — wîX,     m'Y'= — mY,    m'7/= — wiZ,  &c.  : 
eliminating  X'  and  Y'  by  means  of  these  values,  the  first  of 
the  expressions  (360)  becomes 

mYix-x')-mlL{y—y') (361)  : 

but  the  force  whose  components  are  X,  Y,  and  Z  being 
denoted  by  P,  and  the  distance  between  the  points  m  and  m! 
by  /?,  the  cosines  of  the  angles  formed  by  the  direction  of  the 
force  P  with  the  co-ordinate  axes,  will  be  represented  respect- 
ively, by 

x~x'     y—y'    z—z\ 
p  P  p 

and  we  shall  have 

,x-—x      -^ j^y    y      „ -ç^z    z 


X=pl— ^,    Y=P^— ^,    Z=P 


P  p  P 

Substituting  these  values  in  the  expressions  (361),  we  obtain 

mV .- — —{x—x')—mV. iy—y)  ] 

a  quantity  evidently  equal  to  zero. 

In  like  manner,  it  may  be  proved  that  the  other  terms  of 
the  expressions  (360)  destroy  each  other  ;  it  therefore  follows, 
that  when  the  material  particles  m,  rn',  m'\  &.c.  are  subjected 
only  to  their  mutual  attractions,  the  second  members  of  the 
equations  (358)  will  disappear  ;  and  since  this  result  is  inde- 


FREE    MOTION   OF   A    SYSTEM. 


343 


pendent  of  the  position  of  the  origin,  that  point  may  be 
selected  arbitrarily. 

623.   When  either  of   the  three  cases  just    considered 
presents  itself,  the  equations  (358)  will  reduce  to 

dt'  ' 

I,['m(zd^x—xd^z)]_f. 

df^  ' 

2[m{yd^z-zd^p)]_^ 
dp 
The  quantities  included  within  the  brackets  being  exact  dif- 
ferentials, these  equations  may  be  written  under  the  form 
2[*w .  d{xdy — ydx)\_ç. 
~~dP  "  ' 

2[m .  d(zdx — xdz)]  _  „ 

^[m.d{ydz—zdy)]_ 
dt^ 
And  by  multiplying  by  dt,  and  integrating  with  respect  to  the 
time,  denoting  the  arbitrary  constants  by  a,  a',  and  a",  we  shall 

have 

'S\m{xdy—ydx)\=adt    ^ 

'l[m,{zdx—xdz)\=a'dt  \ (362). 

'2[m{ydz  —zdy)'\  =  a"dt  ^ 
624.  To  understand  the  signification  of  these  integrals, 
draw  the  three  rectangular  axes  Ax,  Ay,  and  kz  {Fig.  217), 
and  call  AP=:r,  VQi—y  :  let  AQ,  the  projection  of  the  radius 
vector  Am  on  the  plane  of  x,  y,  be  denoted  by  r,  and  the 
angle  formed  by  AQ  with  the  axis  of  a;  by  tf  ;  the  infinitely 
small  arc  Q,Q,'  described  with  the  radius  r  will  be  expressed 
by  rdè  ;  the  right-angled  triangle  APQ,  gives 

a;=r.costf,    y=r  .svni; 
and,  by  differentiating,  we  obtain 

dx  =  —r  .  sin  6  .d6  +  cos  ê .  dr, 

dy=r .  cos  ê .  dê+sia  ê  .  dr. 

Substituting  these  values  in  the  expression  xdy  —ydx,  we  find 

xdy  -  ydx =r^d6=2x^r'Xrd6=2.a.Te3L  Q  AQ,'  ; 
and  therefore, 


344  DYNAMICS. 

m{xdi/—ydx)=2m{a.re2iCiAQ,'). 
By  forming  similar  products  for  the  other  masses  m',  m",  <kc., 
we  shall  find  that  the  quantity  ^[m{xdi/ —yds)]  is  composed 
of  the  sum  of  the  products  formed  by  multiplying  each  mass 
m,  m',  m",  <fcc.  by  twice  the  area  of  the  elementary  surface 
described  by  the  projection  of  its  radius  vector  Am  on  the 
plane  of  a;,  y,  in  the  time  dt. 

625.  If  we  integrate  again  with  respect  to  the  time,  the 
equations  (362)  will  give 

/l  [m  {xdy — ydx)\  =  at  +  b      ^ 

fi[m{zdx-xdz)]  =  a't  +  b'     \ (363)  ; 

/2[m{ydz-zdy)]  =  a"t  +  b"  ^ 

and  if  the  areas  described  be  supposed  to  commence  from  the 
instant  when  ^=0,  the  constants  6,  6',  and  6"  will  be  equal  to 
zero,  and  the  preceding  equations  will  reduce  to 

fl^ini^xdy—ydxy^—atj 

y  2  [m  {zdx — xdz)\ = «7, 

f^^,n{ydz-zdy)\=a"t. 
These  equations  express  that  the  sums  of  the  products  formed 
by  multiplying  each  mass  by  the  projection  of  the  area  de- 
scribed by  its  radius  vector,  are  constantly  proportional  to 
the  tirn^es  em/ployed  in  describing  these  areas. 

This  enunciation  contains  the  principle  of  the  preservation 
of  areas  in  its  most  general  form. 

626.  The  system  here  considered  has  been  supposed  free  ; 
but  if  it  were  retained  by  a  fixed  point,  the  equations  (358) 
would  only  be  applicable  when  the  origin  was  taken  at  this 
point  :  the  same  may  be  said  of  equations  (363),  which  result 
from  (358).  Thus  the  principle  of  areas  then  becomes  less, 
general,  the  origin  being  no  longer  arbitrary. 

627.  It  has  been  shown  (Arts.  132  and  133)  that  when  the 
system  contains  two  fixed  points,  it  will  be  necessary  to  sat- 
isfy but  one  of  the  general  equations  of  equilibrium  (67). 
The  same  is  true  with  respect  to  equations  (358)  ;  and  there- 
fore but  one  of  the  equations  (362)  will  be  satisfied  :  thus,  the 
principle  of  areas  is  only  true  in  this  case  with  respect  to  one 
of  the  co-ordinate  planes. 

628.  By  comparing  the  results  obtained  in  Art.  155  with 


FREE    MOTION   OF   A    SYSTEM.  S4S 

those  of  Art.  153,  we  shall  find  that  the  quantities  A,  B,  and  C 
represent,  in  Art.  155,  the  sums  of  the  moments  of  the  pro- 
jections of  the  forces  on  the  co-ordinate  planes,  these  mo- 
ments being  taken  with  reference  to  the  origin.  Thus  these 
sums  will  be  the  same  as  those  denoted  by  a,  a',  a"  in  equa- 
tions (362).  Hence,  the  sum  of  the  projections  on  the  prin- 
cipal plane  given  by  equation  (79),  will,  in  the  present  instance, 
be  expressed  by 

/  /  {i:[m{xdy—ydx)'\)  "     (^[m{zdx—xdz)f)  ^     (^[m{ydz—zdy)])  ^  \ 

^  \  dT^  '  dv'  '  dt^  / 

This  expression  may  be  simplified  by  putting  it  under  the 
form 

v'(a" +«"+«'"); 

and  replacing  the  functions  A,  B,  and  C  in  equations  (81)  by 
their  values  a,  a',  and  a",  we  obtain  the  following  expressions 
for  the  cosines  of  the  angles  formed  by  the  principal  plane 
with  the  co-ordinate  planes  : 

a  ^  a' 

cos  a= -, ; — ;— - — r-r->       COS  /3 


a" 
cos  y= . 

The  angles  «,  /s,  y  are  constant  ;  and  hence  we  conclude  that 
the  position  of  the  principal  plane  remains  invariable  during 
the  motions  of  the  several  particles  of  which  the  system  is 
composed. 

General  Principle  of  the  Preservation  of  the  Motion  of  the 
Centre  of  Gravity. 

629.  In  discussing  the  circumstances  of  motion  of  a  sys- 
tem of  material  particles,  acted  upon  by  incessant  forces,  it 
was  proved  that  the  centre  of  gravity  of  the  entire  system 
has  the  same  motion  as  though  the  several  forces  were 
applied  directly  to  that  point.  Thus,  denoting  by  x„  y„  and 
z,  the  variable  co-ordinates  of  the  centre  of  gravity,  we  shall 
have,  as  in  Art.  614, 
MX,=s(mX),    MY,=2(mY),    MZ=2(mZ) (366). 


346  DYNAMICS. 

and, 

'^=^"  ^■=^"  ^'=^ (^«'^- 

630.  If  the  material  points  which  compose  the  system  be 
subjected  only  to  the  action  of  forces  arising  from  their  mutual 
attractions,  the  equations  (367)  will  reduce  to 

^'^/^n     d^y.-a     ^'^/_n. 

these  equations  being  integrated  give 

dx,  dy,     ,      dz, 

—!-=a,     -~=h,     -r^=Cj 
dt  dt  dt 

and  by  a  second  integration  we  find 

x,  =  at-\-a\     y,~bt  +  b',     z=ct-{-c/. 

eliminating  t,  we  have 

x—a'=-{z—c'),    y—h'=-{z—d). 
c  c 

These  equations  appertain  to  a  right  line  in  space,  and  the 

motion  of  the  centre  of  gravity  will  therefore  be  rectilinear. 

This  motion  will  also  be  uniform  ;  for  we  have  the  velocity 

of  the  centre  of  gravity  expressed  by 

y(dxldyldzl\ 

which  is  evidently  a  constant  quantity. 

631.  If  the  masses  m,  m',  m",  (fee.  be  subjected  to  the  action 
of  constant  forces  whose  directions  are  parallel  to  a  given 
line,  we  may  adopt  this  line  as  one  of  the  co-ordinate  axes,  z 
for  example,  and  the  equations  expressing  the  circumstances 
of  motion  of  the  centre  of  gravity,  then  become 

^=0,     ±^1=0,     t^=Z; 
df"       '      dt^       '     dt""         ' 

and  it  may  then  be  proved,  as  in  Arts.  518  and  519,  that  the 
trajectory  described  by  the  centre  of  gravity  is  a  parabola. 

632.  Finally,  it  may  be  shown  that  if  two  or  more  of  the 
bodies  composing  the  system  impinge  against  each  other 
during  the  motion,  the  velocity  of  the  centre  of  gravity  will 
remain  unchanged.  For,  by  the  nature  of  the  centre  of  grav- 
ity, we  have 


FREE   MOTION   OP   A    SYSTEM.  347 

Mx^—l{mx),     My=-S.{my),     M.z,=:L{mz): 
differentiating  with  respect  to  the  time  t^  we  obtain 

M^-l;=^(J-^\     M^l^=^(Jy\     M^-§^=4m^-pi. 
dt        \    dt)'         dt        \    dt  )'         dt        \    dt) 

And  if  we  denote  by  a,  a',  a",  &c.  the  velocities  of  the  parti- 
cles m,  m',  m",  (fee.  before  the  colhsion,  and  by  A,  A',  A",  <fcc. 
the  corresponding  velocities  after  collision,  these  values,  sub- 
stituted in  the  first  of  the  preceding  equations  will  give 

dx 
1(ma)—  the  value  of  M—- '  before  collision, 
dt 

dx 
i:(mA)=  the  value  of  M  — ^after  collision. 
^      ^  dt 

Thus  the  sum  of  the  quantities  of  motion  lost  by  the  impact, 

in  the  direction  of  the  axis  of  x,  will  be  2(ma) — S(mA).     In 

like  manner,  the  sums  of  the  quantities  of  motion  lost  in  the 

direction  of  the  axes  of  1/  and  z  respectively,  will  be 

2(m6)— 2(mB),    and  2(?nc)— 2(mC)  : 

but,  by  the  principle  of  D'Alembert,  these  quantities  should 

maintain  the  system  in  equilibrio  ;  and  we  therefore  have 

2(?/ia)— 2(wA)=0,     2(7n6)  — 2(mB)=0,     2(mc)— 2(wC)=0; 

hence,  the  expressions  — -',  -p,  ~,  which  represent  the  velo- 
dt    dt    dt 

cities  of  the  centre  of  gravity  parallel  to  the  co-ordinate  axes, 
remain  unchanged  by  the  act  of  collision. 

633.  This  property  of  the  centre  of  gravity,  in  virtue  of 
which  its  motion  is  independent  of  the  mutual  actions  of  the 
parts  of  the  system,  constitutes  the  principle  of  the  preserva- 
tion of  the  motion  of  the  centre  of  gravity. 


PART    THIRD. 


HYDROSTATICS. 

OF  THE  PRESSURE  OF  FLUIDS. 

634.  A  fluid  is  a  collection  of  material  particles,  which 
yield  to  the  slightest  eiFort,  and  which  move  freely  among 
each  other  in  all  directions. 

When  the  material  particles  adhere  to  each  other  in  any 
degree,  the  fluid  is  said  to  be  imperfect  ;  in  the  following" 
pages  the  particles  will  be  supposed  entirely  destitute  of  any 
adhesion. 

635.  Fluids  are  divided  into  incompressible  and  compressi- 
ble or  elastic  fluids. 

Incompressible  fluids  are  such  as  always  occupy  the  same 
volume  at  the  same  temperature  ;  such  are  water,  mercury, 
wine,  &c. 

Elastic  fluids  are  those  whose  volumes  admit  of  change  by 
the  application  of  pressure  ;  such  are  the  vapour  of  water, 
atmospheric  air,  and  the  different  gases. 

636.  Let  ABCD  {Fig.  218)  represent  a  vessel  entirely  closed, 
and  filled  with  a  fluid  destitute  of  weight  :  if  two  apertures 
EF  and  HI,  having  equal  surfaces,  be  pierced  in  this  vessel, 
and  if  pistons  K  and  L  be  applied  to  these  apertures,  and  urged 
by  forces  RK  and  SL,  equal  in  intensity,  and  directed  per- 
pendicularly to  the  surfaces  HI  and  EF,  these  forces  will  re- 
main in  equilibrio.  Hence,  it  is  necessary  that  the  pressure 
exerted  upon  the  surface  EF  should  be  communicated  to  the 
surface  HI,  through  the  intervention  of  the  fluid  medium  ] 
and  this  can  only  happen  provided  the  particles  of  the  fluid 
experience  the  same  pressure  at  every  point  of  the  fluid  mass. 
Adopting  the  result  of  this  experiment  as  a  basis,  we  can 

establish  the  following  principle  : 

30 


350  HYDROSTATICS. 

The  characteristic  •property  ofjiuids  is  that  they  transmit 
a  pressure  applied  to  them,  equally  in  all  directions. 

637.  To  express  analytically  this  property,  which  is 
termed  the  principle  of  equal  pressure,  we  shall  consider  a 
fluid  mass  enclosed  in  a  vessel  AL  {Pig.  219)  having  the 
form  of  a  rectangular  parallelopiped,  the  base  of  which  is 
horizontal.  Let  a  piston  be  applied  to  the  upper  surface 
EH  of  the  fluid,  and  let  it  be  urged  downward  by  a  force  P, 
acting:  in  the  vertical  direction  :  the  base  of  the  vessel  will 
experience  the  same  pressure  as  though  the  force  were  ap- 
plied directly  to  it  ;  and  each  portion  of  the  base  will  support 
a  pressure  proportional  to  its  extent  ;  so  that  if  A  denote  the 
area  ABCD,  and  a  the  area  Abed,  of  a  portion  of  this  base  ; 
and  if  p  denote  the  pressure  sustained  by  a,  the  value  of  p 
will  result  from  the  following  proportion, 

A  :  a  :  :  V  :  p. 
Let  a  represent  the  unit  of  surface  ;  we  shall  then  have 

P 

hence,  if  a  represent  the  ratio  between  the  surface  Abed', 
and  the  surface  Abed  assumed  as  the  unit,  the  pressure  P' 
supported  by  the  surface  Ab'c'd,  will  be  expressed  by 

V'=p» (381)  ; 

and  since  all  portions  of  the  fluid  mass  must  sustain  equal 
pressures  for  the  same  extent  of  surface,  it  follows  that  if  the 
surface  containing  «  units  were  situated  in  any  other  portion 
of  the  vessel,  on  the  sides  for  example,  it  would  still  sustain 
the  same  pressure  peo. 

638.  When  the  surface  pressed  is  indefinitely  small,  it  may 
be  represented  by  the  elementary  rectangle  dxdy  ;  and  the 
pressure  exerted  by  the  piston  on  this  elementary  portion  of 
the  surface  of  the  vessel,  will  be  expressed  by  pdxdy  :  this 
expression  will  be  equally  applicable  in  whatever  portion  of 
the  vessel  the  element  may  be  situated,  and  whether  the  sur- 
face be  plane  or  curved. 

639.  In  the  preceding  paragraphs,  the  fluid  has  been  sup- 
posed subjected  merely  to  the  action  of  the  pressure  applied 
at  its  surface  ;  but  when  the  particles  of  the»  fluid  are  acted 


EQUILIBRIUM    OF    FLUIDS.  361 

upon  by  incessant  forces,  the  pressure  will  cease  to  be  con- 
stant throughout  the  mass.  In  this  case,  the  pressure  sus- 
tained by  the  fluid  arises  from  two  distinct  causes  :  1°.  a 
pressure  resuhing  from  the  force  P  apphed  to  the  surface, 
and  equally  distributed  throughout  the  mass  ;  and,  2°.  the 
pressure  arising  from  the  action  of  the  incessant  forces.  The 
latter  pressure  is  usually  different  in  different  parts  of  the 
fluid  mass,  since  each  particle  may  be  acted  on  by  a  force 
having  any  intensity. 

640.  To  offer  an  example  of  this  second  kind  of  pressure, 
let  the  fluid  contahied  in  the  vessel  ABCD  {Fig.  218)  be  con- 
sidered heavy  :  then  we  must  regard  each  particle  as  acted  on 
by  the  force  of  gravity. 

We  shall  find  in  the  sequel,  in  discussing  the  properties  of 
heavy  fluids,  that  the  principle  of  equal  pressure  is  greatly 
modified  by  this  circumstance.  It  follows  from  the  preceding 
remarks,  that  the  pressure  p  should  in  general  be  regarded  as 
variable,  in  passing  from  one  point  to  another  of  a  fluid  mass, 
when  the  particles  are  acted  upon  by  incessant  forces.  In 
this  case,  the  pressure  p  exerted  at  any  point  whose  co-ordi- 
nates are  ar,  y,  z,  when  referred  to  the  unit  of  surface,  must 
be  understood  to  denote  the  pressure  which  would  be  exerted 
upon  a  unit  of  surface,  if  every  point  in  this  unit  should  sus- 
tain a  pressure  equal  to  that  exerted  at  the  point  x,  y,  z. 

General  Equations  of  the  Equilihriinn  of  Fluids. 

641.  Let  a  fluid  particle  solicited  by  incessant  forces  be 
supposed  to  rest  in  equilibrio  in  a  fluid  mass,  and  let  it  be 
required  to  determine  the  equations  necessary  to  establish  the 
state  of  equilibrium. 

For  this  purpose,  let  the  co-ordinate  plane  of  x,  y  be 
assumed  horizontal  and  above  the  fluid  mass,  which  we  will 
suppose  divided  into  infinitely  small  rectangular  parallelo- 
pipeds  by  planes  parallel  to  the  co-ordinate  planes.  Let  dM. 
represent  one  of  these  elements  whose  co-ordinates  are  x^  y,  and 
z  :  the  volume  of  this  element  will  be  expressed  by  dxdydz  ; 
and  by  multiplying  this  volume  by  the  density  D,  supposed 
constant  throughout  the  element,  we  shall  have  D .  dxdydz 


352  HYDROSTATICS. 

for  the  expression  of  the  elementary  mass  of  the  fluid  :  hence, 
we  derive  the  equation 

cm=B .  d.Td}/dz (382). 

If  X,  Y,  and  Z  represent  the  incessant  forces  which  act  upon 
the  element  dM,  and  which  are  supposed  constant  throughout 
the  extent  of  this  element,  X^^M,  YdM,  and  ZrfM  will  express 
the  moving  forces  exerted  upon  the  elementary  parallelopiped, 
and  these  forces,  acting  conjointly  with  the  pressure  sus- 
tained by  the  several  faces  of  the  element,  should  maintain 
this  element  in  equilifcrio.  Let  the  superior  surface  d.idi/  of 
the  parallelopiped  be  extended  (Pig.  220)  until  its  area 
becomes  equal  to  the  assumed  unit  represented  by  BG  ;  and 
let  the  pressure  p  sustained  throughout  this  unit  be  con- 
ceived constant,  and  equal  to  that  exerted  at  each  point  of  the 
face  axdy.  When  the  ordi'^ate  BT)=z  is  changed  into 
D^—z  +  dZj  the  pressure  p,  which  varies  with  z,  will  become 

and  will  express  the  pressure  exerted  on  the  unit  of  surface, 
each  point  of  which  sustains  a  pressure  equal  to  that  sup- 
ported by  the  points  in  the  base  EF  of  the  parallelopiped. 
Consequently,  to  obtain  the  total  pressures  on  the  superior 
base  BG  and  on  the  inferior  base  EF  of  the  element,  we 
must  multiply  the  surfaces  BG  and  EF,  each  of  which  is 
equal  to  dxdy,  by  the  respective  pressures  exerted  upon  their 
unit  of  surface  :  thus,  we  shall  obtain  for  the  pressures  sup- 
ported by  BG  and  EF, 

pdxdy^     and  (ji-\--~dz)dxdy  ] 

the  first  of  these  pressures  is  exerted  downwards,  and  the 
latter  upwards.  Their  difference  will  be  a  pressure  exerted 
upwards,  if  we  suppose  the  pressure  to  increase  with  the 
co-ordinate  z,  and  it  will  be  expressed  by 

-^dzdxdy  : 

and  since  this  difference  should  sustain  in  equilibrio  the  ver- 
tical force  Z^iM,  we  shall  have 


EaUILIBRIUM   OF   FLUIDS.  363 

^dzdxdy=ZdM.  : 
dz  ^ 

substituting  for  dM  its  value  given  by  equation  (382),  and 
reducing,  we  find 

dz 
In  like  manner,  by  denoting  the  lateral  pressures  on  a  unit 
of  surface  exerted  against  the  faces  dxdz,  dydz,  by  q  and  r, 
we  shall  obtain 

—  =:DY  =DX. 

dy  ^     dx 

It  has  been  shown  (Art.  640)  that  the  pressures  exerted  upon 
any  one  of  the  faces  is  composed  of  the  pressure  uniformly 
distributed  throughout  the  fluid,  and  of  the  pressure  due  to 
the  incessant  forces.  Thus,  to  estimate  the  pressure  qdxdz, 
exerted  upon  the  face  dxdz,  it  is  obvious  that  this  pressure 
•  may  be  considered  as  resulting  from,  1°.  The  pressure  exerted 
upon  the  superior  base,  which  is  transmitted  equally  through- 
out the  parallelopiped  ;  and,  2°.  The  pressure  due  to  the 
incessant  forces  exerted  upon  the  particles  which  compose 
the  parallelopiped.  But  the  pressure  exerted  upon  the  upper 
base  being  pdxdy,  it  will  be  transmitted  to  the  face  dxdz, 
exerting  a  pressure  pdxdz  proportional  to  the  area  of  this 
face. 

The  incessant  forces  being  Xé?M,  Yc?M,  and  XdM.  respect- 
ively, the  pressure  arising  from  their  joint  action  will  be  a 
function  of  their  intensities,  which  we  shall  represent  by 

F(X^M,  Ydm,  ZdM)  ; 
and  we  shall  thus  obtain 

qdxdz=pdxdz  +  F(X.dm,  Ydm,  Zdm) (383). 

The  function  represented  by  • 

F(Xrfw,  Ydm,  Zdm) 
must  be  such  that  it  will  disappear  when  the  forces  are  sup- 
posed equal  to  zero  :  hence  it  is  necessary  that  every  term  of 
the  function  should  contain  at  least  one  of  the  factors  X^M, 
YdM,  or  ZdM.  :  and  by  arranging  the  terms  with  reference 
to  the  powers  of  dM,  commencing  with  the  least,  we  may 
suppose 

Z 


354  HYDROSTATICS. 

F(X^M,  YdM,  ZdM)^hX<m  +  J<iY(M  +  VZdM+&,c. 
Substituting  this  value  in  equation  (383),  we  shall  have 

qdsdz=pd3;dz  +  hX(M+l^Y(m-^VZ(m-\-&.c.  ■ 
and  replacing  dM  by  its  equal  Ddxd^dz,  this  equation  will 
become 
qdxdz  =pdxdz  +  T)hKdxdydz + jy^Ydxdydz 

-{-BFZdxdydz  +  âcc: 
<iividing  by  dxdz,  there  results 

q=p-\-'DhXdy+BNYd7/  +  T)PZdi/-]-6cc (384). 

The  terms  BLXdy,  DNYrfy,  BPZdy,  &c.  being  infinitely 
small  with  respect  to  p,  it  follows  that  the  equation  (384) 
may  be  reduced  to 

q=p. 
In  like  manner,  it  may  be  demonstrated  that  r=p  ;   and 
hence,  the  equations  of  equilibrium  will  become 

$=DZ,     i^=DY,     i^=DX (385). 

dz  dy  dx 

If  we  multiply  these  equations  by  dz,  dy,  and  dx  respectively, 
and  take  their  sum.  we  shall  obtain  an  expression  for  the 
diâërential  of  the  pressure,  when  the  co-ordinates  x,  y,  and 
z  are  supposed  to  vary  together  ;  thus, 

dp=^dx^'^-^dy+^dz=B(Xdx+Ydy+Zdz) (386). 

dx         dy         dz 

Such  is  the  equation  which,  when  integrated,  will  determine 

the  pressure  upon  the  unit  of  surface  at  any  point  of  the 

fluid. 

Application  of  the  General  Equations  of  Equilibrium  to 
Incompressible  Fluids. 

642.  Let  us  suppose  an  incompressible  homogeneous  fluid 
to  be  in  equilibrio  in  a  vessel  capable  of  opposing  an  indefi- 
nite resistance  to  pressure  :  the  pressure  p  exerted  upon  the 
unit  of  surface,  at  a  point  whose  co-ordinates  are  x=a,  y=b, 
z=c,  will  be  determined  by  substituting  the  values  a,  b,  and  c 
for  X,  y,  and  z,  in  the  integral  of  equation  (386)  :  and  if  the 
density  D  be  supposed  constant,  the  determination  of  the 


INCOMPRESSIBLE    FLUIDS.  355 

value  of  p  will  depend  on  the  possibility  of  integrating  the 
formula 

Xdx  +  Ydy  +  Zdz (387). 

This  integration  will  always  be  possible,  when  the  pre- 
ceding expression  is  an  exact  differential  of  the  variables 
X,  y,  and  z. 

Let  it  be  supposed  that  this  condition  is  fulfilled,  and  that 
the  pressure  at  any  point  on  the  sides  or  bottom  of  the  vessel 
has  been  determined  ;  this  pressure  will  be  destroyed  by  the 
resistance  of  the  vessel.  But  if  we  consider  a  point  in  the  free 
surface  of  the  fluid,  and  suppose  that  no  exterior  pressure  is 
applied  to  the  fluid  by  means  of  a  piston  or  otherwise,  it  is 
obvious  that  the  pressure  at  such  point  will  be  equal  to  zero. 
The  same  being  true  for  every  point  in  the  free  surface  of  the 
fluid,  it  follows  that  in  passing  from  any  point  in  the  surface 
of  the  fluid  to  a  consecutive  point  in  the  same  surface,  the 
pressure  j9  will  remain  invariable,  being  equal  to  zero  at  each 
of  these  points  ;  hence  dp=0,  and  the  equation  (38G)  con- 
dered  as  applicable  to  points  situated  in  this  surface,  will 
reduce  to 

Xda;  +  Ydi/+Zdz=0 (388). 

This  equation  will  likewise  appertain  to  the  surface  of  the 
fluid  when  this  surface  experiences  a  constant  pressure,  that  of 
the  atmosphere  for  example,  since  we  shall  still  have  dp=0. 

It  will  also  subsist  for  those  points  within  the  fluid  mass 
which  su&tain  equal  pressures. 

643.  When  the  expression  (387)  is  an  exact  diflèrential, 
and  the  equation  (388)  is  satisfied,  we  shall  have  ijo=0,  and 
the  pressure,  if  it  exist,  must  be  constant.  But,  in  this  case, 
in  order  that  the  equilibrium  may  be  preserved,  it  is  neces- 
sary that  the  resultant  of  the  forces  exerted  upon  each  par- 
ticle in  the  surface,  and  directed  towards  the  interior  of  the 
fluid,  should  be  normal  to  the  surface  of  the  fluid  :  for  if  it 
were  not,  we  might  decompose  this  resultant  into  two  forces, 
one  normal  and  the  other  tangent  to  the  surface  ;  and  it  is 
evident  that  the  latter  would  impart  a  motion  to  the  fluid  par- 
ticle. 

644.  This  condition  is  likewise  indicated  by  the  equation 

Z  2 


358  HYDROSTATICS. 

(388)  ;  for,  let  x\  y\  and  z'  represent  the  co-ordinates  of  a  par- 
ticle in  the  surface  of  the  fluid,  and  X,  Y,  and  Z  the  incessant 
forces  applied  to  this  particle.  The  general  equations  of  the 
normal  to  a  curved  surface  at  the  point  x\  y\  z,  are 


x—x  — ■j—{z—z') 

ax 

y—v  =  — ; — {z—z) 

^     ^  dy'  ' 


(389); 


dz' 
and  if  we  substitute  in  these  equations  the  values  of  -— ^and 

CJL3j 

dz' 

-—  determined  by  equation  (388),  the  equations  (389)  will 

become  those  of  the  normal  to  the  surface  of  which  (388)  is 
the  equation.  But  by  regarding  X,  Y,  and  Z  as  functions 
of  the  co-ordinates  x^  y,  and '2:,  and  employing  the  usual  nota- 
tion, the  equation  (388)  will  give 

~  dx'~~7l'     ~  dy'~  Z' 
Substituting  these  values  in  (389),  we  find,  for  the  equations 
of  the  normal  at  the  point  x\  y\  z\ 

x-x'=^{z-z'),    y-y'=^iz-z'). 

These  equations  are  precisely  similar  to  those  of  the  result- 
ant cf  the  forces  X,  Y,  and  Z,  found  in  Art  57. 

645.  The  equation  (388),  when  susceptible  of  being  inte- 
grated, leads  to  several  remarkable  consequences.  For,  if  we 
represent  the  integral  of  this  equation  by  F{x,  y,  z)-{-C,  and 
make  C  =  — A,  we  shall  have 

F{x,y,  z)=A. 
If  we  assign  to  A  arbitrary  values  successively  increasing, 
such  as  0,  a,  a',  a",  (fee,  we  shall  obtain  the  equations 

F(x,y,z)=0, 

F{x,y,z)=a, 

F{x,y,  z)=a', 

F{x,  y,  z)=a", 


F{x,  y,  z)=a^"\ 
<fec.        (fee. 


INCOMPRESSIBLE    FLUIDS.  357 

Each  of  these  equations  being  diflferentiated  will  produce 
equation  (388),  and  among  them  will  be  found  that  apper- 
taining to  the  surface  of  the  fluid,  which  is  supposed  to  have 
produced  equation  (388)  by  differentiation. 

Let  this  equation  be  represented  by  F{x,  y,  2;)=a^"'  :  then 
the  other  equations  will  appertain  to  different  surfaces,  each 
of  which  will  possess  the  property,  that  the  resultant  R  of  the 
forces  X,  Y,  Z,  exerted  upon  any  particle  situated  in  such  sur- 
face, will  be  perpendicular  to  the  surface. 

The  directions  of  the  forces  being  cut  perpendicularly  by 
the  surfaces  of  constant  pressure,  such  surfaces  are  said  to  be 
level.  If  we  suppose  the  arbitrary  constants  0,  a,  a',  of',  &c. 
to  differ  by  indefinitely  small  increments,  the  fluid  mass  will 
be  divided  by  these  level  surfaces  into  a  series  of  extremely 
thin  layers,  which  are  denominated  level  strata. 

646.  It  follows,  from  the  preceding  remarks,  that  when  the 
particles  of  the  fluid  are  solicited  by  forces  constantly  di- 
rected towards  a  fixed  point,  its  exterior  will  assume  the 
spherical  form.  The  same  consequence  may  be  deduced  ana- 
lytically. For,  let  the  origin  be  taken  at  the  centre  of  attrac- 
tion, and  denote  by  .v,  y,  z  the  co-ordinates  of  a  particle  dM.  in 
the  surface  of  the  fluid:  the  distance  of  the  point  t,  i/,z  from  the 
origin  will  be  expressed  by  ^{x^  +f/^  -\-z^).  If  this  distance 
be  denoted  by  r,  and  the  force  of  attraction  exerted  upon  the 
particle  dM  by  a,  the  cosines  of  the  angles  formed  by  the  direc- 
tion of  this  force  with  the  co-ordinate  axes  will  be  expressed 

by—)  ->  ^^^  -  j  3.nd  the  components  of  the  force  a  will  be 


r    r 


X=-A-,     Y=-x^,     Z=-A-; 
r  r  r 

the  negative  signs  are  prefixed  to  these  components  because 
they  tend  to  diminish  the  co-ordinates  of  the  particle  dM.. 
By  substituting  these  values  in  equation  (388),  we  shall 
obtain  for  the  diîîerential  equation  of  the  surface  of  the  fluid 

t(xdx+ydy  +  zdz)=0 (390). 

r 

Suppressing  the  common  factor    ,  and  integrating,  we  find 


358  HYDROSTATICS. 

an  equation  appertaining  to  a  spherical  surface  ;  hence  the 
surface  of  the  fluid  will  be  spherical. 

647.  If  the  radius  of  the  sphere  be  very  great  in  compari- 
son with  the  extent  of  the  surface,  as  is  the  case  when  we 
consider  a  small  portion  of  the  earth's  surface,  the  curvature 
will  be  insensible,  and  the  surface  may  therefore  be  regarded 
as  a  plane. 

648.  The  integration  of  equation  (390)  was  effected  imme- 
diately in  consequence  of  equation  (388)  becoming,  in  that 
example,  a  particular  case  of  the  theorem  demonstrated  in 
Art.  436,  relative  to  forces  directed  to  fixed  centres.  It  is  by 
virtue  of  this  theorem  that  equation  (388)  will  always  be 
integrable  in  such  cases  as  refer  to  the  equilibrium  of  fluids 
resting  upon  fixed  surfaces. 

649.  If,  in  equation  (386),  we  replace  the  quantity  within 
the  brackets  by  its  equal  d[F{x,  y,  z)],  we  shall  obtain 

dp=T>xd[F{x,y,z)]] 
or,  by  division, 

^=4F(:r,  i/,z)] (391). 

But  d[F{x,  y,  z)]  being  by  hypothesis  an  exact  differential, 
-^  must  likewise  be  an  exact  differential  ;  hence,  D  will  con- 
tain no  variable  except/?  ;  this  condition  may  be  expressed  by 
the  equation 

Ty^fp (392). 

If  the  pressure  ^  be  supposed  constant,  the  density  D  will 
be  likewise  constant,  and  (391)  will  reduce  to 
d[F{x,7/,z)]=0. 

The  integration  of  this  equation  will  reproduce  that  already 
found  in  Art.  645,  the  properties  of  whic^  have  been  dis- 
cussed. 

650.  The  fluid  being  still  supposed  incompressible,  but 
heterogeneous,  the  density  D  will  be  variable  ;  and  in  order 
that  the  pressure  p  may  be  determinate,  the  quantity 
DQidx+Ydy+Zdz)  must  be  an  exact  differential  :  but     if 


ELASTIC    FLUIDS.  359 

X-dx -\-Y dy  ^'Zadz  be  likewise  supposed  an  exact  differential, 
it  will  appear,  as  in  equation  (392),  that  the  density  will  be 
always  a  function  of  the  pressure.  Thus  the  pressure  and 
density  will  become  constant  together,  and  will  remain  in- 
variable for  all  points  situated  in  a  level  stratum. 

We  conclude,  therefore,  that  a  heterogeneous  fluid  mass 
cannot  remain  in  equilibrio,  unless  it  be  disposed  in  such  man- 
ner that  each  of  the  level  strata  shall  be  of  equal  density 
throughout.  The  law  of  variation  in  the  density  in  passing 
from  one  stratum  to  another,  will  depend  on  the  manner  in 
which  D  is  expressed  in  functions  of  x^  y,  and  z  :  and  since 
the  nature  of  the  function  is  entirely  arbitrary,  the  law  of 
the  density  will  likewise  be  arbitrary. 

Application  of  the  General  Equations  of  Equilibrium  to 
Elastic  Fluids. 

651.  The  characteristic  property  of  an  elastic  fluid  is  its 
power  of  sustaining  compression,  and  subsequently  regaining 
its  original  volume  and  elasticity,  when  the  compressing  force 
is  removed. 

Thus,  a  fluid  which  is  elastic  exerts  in  addition  to  the 
pressure  due  to  the  forces  which  act  upon  it,  an  eflîbrt  arising 
from  the  elasticity  of  its  particles. 

It  has  been  ascertained  experimentally,  that  this  effort, 
which  is  called  the  elastic  force  of  the  fluid,  is  proportional  to 
its  density,  so  long  as  the  temperature  remains  invariable. 

Thus,  if  .we  suppose  the  temperature  to  remain  constant, 
and  represent  by  P  that  pressure  exerted  upon  the  unit  of 
surface  which  is  necessary  to  produce  a  certain  density 
assumed  as  the  unit,  this  density  will  be  doubled  when  the 
pressure  becomes  2P  ;  trebled  when  the  pressure  becomes 
3P,  (fee.  ;  and  hence,  if  the  density  be  expressed  by  D,  the 
corresponding  pressure  will  be  PD.  This  pressure  being 
denoted  by  p,  we  shall  have 

p^VJy (393)  ; 

the  quantity  p  represents,  as  heretofore,  the  pressure  exerted 
upon  the  unit  of  surface. 


360  HYDROSTATICS. 

653.  By  combining  equation  (393)  with  the  equation 
dp=\y{lidx^-Ydy+Zdz\ 
there  results 

dp  _K.dx-\-Yd'y-\-'Ldz  tOQA\ 

7 p ^"^^^^  ' 

and  by  integration,  we  have 

_PlLdx  +  Ydy-\-Zdzç, 

653.  The  temperature  being  supposed  constant  throughout 
the  mass,  and  the  nature  of  the  fluid  particles  everywhere 
the  same,  the  quantity  P  will  be  constant,  and  may  therefore 
be  placed  without  the  integral  sign  :  thus,  by  representing 
the  constant  C  by  log  C,  we  shall  have 

or,  if  we  denote  by  e  the  base  of  the  Naperian  system,  this 
equation  will  reduce  to 

fÇ%.dx+Ydy+Zdz) 

log^=loge  ^  +log  C  : 

reducing,  and  passing  from  logarithms  to  numbers,  we  find 

/(Xrfj+Yrfy+Zdz) 

p=C'e  P 

This  value  being  substituted  in  equation  (393),  we  obtain 

/(Xrfj+Yrfy+Zdz) 

Ce  ^ 


P 

The  pressure  and  density  being  both  functions  of  the  quan- 
tity/(Xc^:r+Yc?y+Z^2;),  they  will  become  constant  at  the 
same  time  ;  and  hence,  the  density  of  the  fluid  throughout 
each  level  stratum  will  remain  invariable.  The  value  of  the 
density  in  any  stratum  results  immediately  from  the  pre- 
ceding equation. 

654.  It  should  be  remarked,  that  in  the  case  of  elastic 
fluids,  the  equation 

:Ldx+Ydy+Zdz=Q 

cannot  be  deduced  from  the  hypothesis  of  ^=0:  for,  if  we 
suppose  ^=0,  the  equation  will  give  D=0  ;  and  hence,  we 
perceive  that  it  would  be  necessary  that  the  density  of  the 


PRESSURE    OF    HEAVY    FLUIDS.  361 

fluid  should  be  likewise  equal  to  zero  ;  a  supposition  which 
would  destroy  the  existence  of  the  fluid. 

We  conclude,  therefore,  that  in  an  elastic  fluid,  the  pressure 
cannot  be  equal  to  zero  at  the  surface  ')f  the  fluid,  as  is  the 
case  with  incompressible  fluids.  Thus,  a  mass  of  elastic 
fluid  cannot  be  in  equilibrio  unless  contained  in  a  close 
vessel,  or  extended  indefinitely  in  space. 

Of  the  Pressure  of  Heavy  Fluids, 

655.  It  is  now  proposed  to  examine  the  circumstances  of 
equilibrium  in  fluids  whose  particles  are  acted  on  by  the  force 
of  gravity.  For  this  purpose,  let  it  be  supposed  that  a  vessel 
is  placed  upon  a  horizontal  plane,  and  filled  with  water,  or 
other  heavy  fluid,  to  a  certain  height.  The  surface  of  the 
fluid,  as  has  been  demonstrated,  will  assume  a  horizontal  posi- 
tion ;  let  this  surface  be  assumed  as  the  plane  of  a-,  y,  and  let 
the  co-ordinates  z  be  reckoned  positive  downwards  ;  the  force 
of  gravity  being  the  only  force  exerted  upon  the  fluid  parti- 
cles, we  shall  have 

X=0,     Y=0,     Z=^; 

and  the  equation  (386)  will  become 

dp^Dgdz. 

The  density  of  the  fluid  and  the  intensity  of  gravity  being 

supposed  constant,  the  integration  of  this  equation  will  ^ive 

p='Dgz-\-C (395). 

To  determine  the  value  of  the  constant  C,  we  make  z—0, 
and  since  the  pressure  p  is  equal  to  zero  at  ihe  same  time,  we 
deduce  C=0  :  thus  the  equation  (395)  is  reduced  to 
p=T>gz (396). 

656.  If  a  horizontal  plane  be  drawn  below  the  surface  of 
the  fluid,  every  point  in  such  plane  will  have  a  common  ordi- 
nate z  ;  and  the  pressure  p—Dgz  will  therefore  be  constant 
throughout  this  plane. 

657.  Let  h  represent  the  distance  between  the  surface  of 
the  fluid  and  the  horizontal  base  of  the  vessel  ;  the  pressure 
supported  by  the  unit  of  surface  of  the  base  will  be  determined 


362 


HYDROSTATICS. 


by  equation  (396),  in  which  we  replace  z  by  A,  and  thus 
obtain 

p=T)gh (397). 

Let  J)'  represent  the  pressure  supported  by  the  entire  base, 
which  is  supposed  to  contain  h  units  of  surface  :  the  quantity 
p  will  be  contained  b  times  in  p'  :  we  therefore  have 

p'=bp (398), 

and  by  substituting  for  p  its  value  given  in  equation  (397), 
we  find 

p'=Tighb (399). 

But  bh  represents  the  volume  of  a  prism  whose  base  is  6, 
and  height  h  ;  and  by  multiplying  this  vokime  by  the  density 
D,  we  obtain  bliD  for  the  mass  of  the  prism  :  therefore  bg/iD 
will  express  the  weight  of  such  prism  ;  and  hence,  it  appears 
that  the  base  b  supports  a  pressure  equal  to  the  weight  of  the 
column  of  fluid  which  rests  immediately  upon  it. 

658.  The  pressure  p',  exerted  by  the  same  fluid,  being 
dependent  only  on  the  base  b  and  height  h,  it  follows  that  the 
pressures  supported  by  the  bases  of  diflerent  vessels  will  be 
equal,  whatever  may  be  the  forms  of  the  vessels,  provided 
their  bases,  and  the  heights  of  the  fluid  above  them,  be 
respectively  equal. 

659.  To  determine  the  lateral  pressure  exerted  against  the 
sides  of  the  vessel,  let  da  represent  the  element  of  this  sur- 
face, and  z  the  distance  of  the  element  from  the  surface  of  the 
fluid  ;  the  pressure  j)  (referred  to  the  unit  of  surface),  which 
is  supported  by  the  element  da,  will  be  determined  by  equa- 
tion (396)  :  this  value  being  substituted  in  equation  (399), 
and  the  area  b  being  replaced  by  da,  we  obtain  T) .  gz .  da>  for 
the  expression  of  the  entire  pressure  on  the  element  da.  A 
similar  expression  may  be  obtained  for  the  pressure  upon 
each  element  ;  and  since  the  pressures  will  be  exerted  in  par- 
allel directions  when  the  side  of  the  vessel  is  supposed  plane, 
we  shall  have,  for  the  total  pressure  exerted  against  the  side, 

p'=fDgzda. 
The  second  member  of  this  equation  contains  two  variables, 
one  of  which  must  be  eliminated  before  the  integration  can 


PRESSURE   OF    HEAVY   FLUIDS.  363 

be  efiected.     This  elimination  is  readily  accomplished  when 
the  figure  of  the  surface  <y  is  known. 

660.  Let  it  be  required,  for  example,  to  determine  the  pres- 
sure exerted  against  the  inclined  rectangle  ACDB  {Fig.  221), 
whose  sides  AB  and  CD  are  parallel  to  the  horizon.  Denote 
by  h  and  I  the  base  AB  and  length  BD  of  the  rectangle,  and 
conceive  its  surface  to  be  divided  into  an  infinite  number  of 
elements,  by  lines  parallel  to  AB  or  CD  ;  the  pressure  will  be 
the  same  upon  every  point  of  the  same  element.  Let  v  denote 
the  distance  D/  of  any  one  element  af  from  the  base  CD  ; 
the  height  of  this  element  will  be  expressed  by  dv=ae^  and 
the  surface  of  the  element  by 

abXae=bdv  ; 
substituting  this  value  for  do  in  the  expression  /Dg-zda,  we 
obtain 

/Dgzdu^/Dgzbdv  : 
such  will  be  the  expression  for  the  pressure  exerted  upon  the 
surface  ABDC.  The  integral  should  be  taken  between  the 
limits  v—0  and  v=l,  the  variable  z  being  previously  elimi- 
nated. To  effect  this  elimination,  let  (p  represent  the  angle 
BDL  included  between  the  plane  of  the  rectangle  and  the 
vertical  line  NL,  and  a  the  distance  DN  of  the  superior  base 
CD  from  the  surface  of  the  fluid  ;  we  shall  have 

K/orLN=NDH-DL; 
or, 

z=a  +  v .  cos  <J)  : 

hence,  the  pressure  exerted  upon  the  surface  will  be  ex- 
pressed by 

p'=f'Dg{a-{-v  .  cos  <p)bdv  ; 
and  by  performing  the  integration  indicated,  we  find 

p'='Dgb{av-\-^v''  cos(|>)4-C. 
The  integral  being  taken  between  the  limits  v=0,  and  v=l, 
we  obtain 

p'=T>gb(al-{-ll^  cos  (p). 

661.  To  determine  the  point  of  application  of  the  resultant 
of  all  the  pressures  exerted  upon  the  rectangle,  we  remark, 
in  the  first  place,  that  this  point  must  be  situated  upon  the  line 


364  HYDROSTATICS. 

EH,  which  bisects  the  sides  AB  and  CD.  We  next  regard 
the  pressures  exerted  upon  the  different  points  of  the  surface 
ABDC  as  parallel  forces,  and  determine  their  moments  with 
reference  to  a  vertical  plane  passing  tlirough  the  horizontal 
line  CD  :  the  pressure  sustained  by  the  element  abfe  being 
Dgzbdv,  its  moment  will  be  expressed  by  DgzbdvXv  .  sin  (p  ; 
and  by  denoting  the  distance  EG  of  the  point  of  application 
of  the  resultant  from  the  line  CD  by  v,,  the  principle  of  mo- 
ments will  give 

p'v,  sin  4)= sin  <pfDgzhvdv  ; 

or, 

*"  p'v,—fT)gzhvdv. 

If,  in  this  expression,  we  replace  z  by  its  value  determined  in 
the  preceding  Art.,  we  shall  obtain 

v'v,—Y)ghf{avdv-\-cos  <p .  v^dv)  ; 
whence^  by  integration. 


— +cos^yJ+C. 


The  integral  being  taken  between  the  limits  v==0  and  v=^, 
there  results 


p'v=Dgh  ^^-^+008  Ç-J  ; 


and  by  substituting  for  p'  its  value,  we  find,  after  reduction, 

al  ,  l^ 

2+cos<.- 

v=-  J- 

a  +  cosç>  - 

Having  found  the  pressures  exerted  upon  the  base  and 
upon  each  of  the  sides  of  the  vessel,  we  combine  these  pres- 
sures, and  determine  their  resultant  :  such  resultant  will 
express  the  entire  pressure  produced  by  ihe  fluid. 

662.  We  will  next  consider  a  body  immersed  in  a  homo- 
geneous heavy  fluid  :  the  pressure  exerted  by  this  fluid 
against  any  portion  of  the  surface  of  the  body  may  be  deter- 
mined by  the  method  for  finding  the  pressure  against  the 
sides  of  a  vessel  ;  but  when  it  is  required  to  consider  the 
total  pressure  exerted  against  tiie  surface  of  a  body  immersed 


PRESSURE    OF   HEAVY   FLUIDS.  365 

in  a  fluid,  we  commonly  employ  the  following  theorems,  the 
truth  of  which  will  be  demonstrated. 

1°.  The  pressures  exerted  upon  the  surface  of  a  body  en- 
tirely immersed  in  a  fluid  have  a  single  resultant  ^  which  is 
vertical  and  directed  ujjwards. 

2°.  The  resultant  of  all  the  pressures  is  equal  in  intensity 
to  the  weight  of  the  fluid  displaced. 

3°.  The  line  of  direction  of  this  resultant  passes  through 
the  centre  of  gravity  of  the  displaced  fluid. 

4°.    The  horizontal  pressures  destroy  each  other. 

To  establish  the  truth  of  these  propositions,  let  us  suppose 
a  vessel  ADE  {Fig.  222)  to  be  filled  with  a  heavy  fluid  in 
equilibrio,  and  let  a  portion  of  this  fluid  KL  be  conceived  to 
become  solid,  its  density  remaining  unchanged  :  the  state  of 
equilibrium  will  not  be  disturbed  by  this  change.  But  this 
solid  is  urged  downwards  by  a  force  equal  to  its  weight, 
applied  at  its  centre  of  gravity.  This  force  can  only  be 
destroyed  by  the  resultant  of  all  the  pressures  exerted  by  the 
fluid  against  the  solid  ;  hence,  it  follows  that  these  pressures 
must  have  a  single  resultant  equal  in  intensity  to  the  weight 
of  the  displaced  fluid,  and  that  this  resultant  must  be  applied 
at  the  centre  of  gravity  of  the  displaced  fluid,  and  be  directed 
vertically  upwards.  Moreover,  as  the  direction  of  the  result- 
ant is  vertical,  the  horizontal  pressures  will  mutually  destroy 
each  other. 

When  a  body  is  partially  immersed  in  a  fluid,  an  equilibrium 
cannot  subsist  unless  the  centres  of  gravity  of  the  body  and 
of  the  fluid  displaced  be  situated  upon  the  same  vertical  line  : 
this  condition  will  necessarily  be  fulfilled  when  the  body  is 
entirely  immersed,  provided  it  be  homogeneous  ;  since  its 
centre  of  gravity  will  then  coincide  with  that  of  the  fluid 
displaced. 

The  buoyant  eflbrt  exerted  by  the  fluid  being  directed  along 
a  line  which  passes  through  the  centre  of  gravity  of  the  dis- 
placed fluid,  that  point  is  called  the  centre  of  buoyancy. 

663.  Let  V  represent  the  volume  of  fluid  displaced,  and  v' 
that  of  the  body  ;  D  the  density  of  the  fluid,  and  D'  that  of 
the  body  :  the  weights  of  the  volume  of  displaced  fluid,  and 


366 


HYDROSTATICS. 


of  the  body  will  be  respectively  Dgv  and  D'gv'.    If  the  body 
be  supposed  to  rest  in  equilibrio,  we  shall  have 

Dgv=:D'g-v'  ] 
and  if  we  suppose  it  to  be  entirely  immersed,  the  volumes  v 
and  v'  will  be  equal,  and  the  densities  D  and  D'  must  likewise 
be  equal,  in  order  that  tft«^  equilibrium  may  be  preserved. 

But  if  the  weight  of  the  body  bc.lelsg  than  that  of  the  fluid 
displaced,  w^e  shall  have  ■*>..    .. 

and  the  body  will  be  urged  upwards  by  a  force  equal  to  the 
difference  Dgp—'D'gi'. 

If,  on  the  contrary,  we  should  have 
T>gv<D'gv', 
the  body  would  tend  downwards   with  a  force  equal  to 
Wgv' — T>gv. 

Of  the  Equilihrium,  Stability,  and  Oscillations  of  Floating 

Bodies. 

664.  The  propositions  demonstrated  in  Arts.  662  and  663 
establish  two  principles  which  serve  as  tlie  basis  of  the  theory 
of  floating  bodies  ;  these  principles  are, 

l'^.  When  a  body  is  partially  or  totally  miTnersed  in  a 
fluid,  an  equilibrium  cannot  subsist  unless  the  centre  of 
gravity  and  centre  of  buoyancy  be  situated  upon  the  same 
vertical  line. 

2°.  If  an  equilibrimn  be  maintained,  the  weight  of  the 
body  will  be  equal  to  that  of  the  fluid  displacef. 

The  latter  principle  is  frequently  employed  for  the  purpose 
of  estimating  the  weight  of  a  ship  either  with  or  without  her 
cargo.  For  this  purpose,  we  measure  the  capacity  of  the 
part  immersed,  and  allow  a  weight  of  one  ton  for  every  35 
cubic  feet  which  it  contains.  By  taking  the  difference  of  the 
weights  of  the  vessel  with  and  without  the  cargo,  the  weight 
of  the  latter  may  be  obtained.  We  can  also  arrive  at  the 
same  result,  by  simply  measuring  the  additional  portion  of  the 
vessel  immersed,  when  the  cargo  is  introduced. 


EQUILIBRIUM    OF   FLOATING   BODIES.  367 

665.  The  horizontal  surface  of  the  fluid  is  called  the  flane 
of  floatation. 

666.  If  V  denote  the  volume  of  fluid  displaced,  D  its  den- 
sity, and  g  the  intensity  of  the  force  of  gravity,  the  weight  P 
of  the  body  ABC  {Fig.  223),  which  floats  upon  the  surface 
of  the  fluid,  and  is  partially  immersed,  will  be  equal  to  Dgv. 

667.  When  the  floating  body  and  fluid  are  both  homogene- 
ous, the  centre  of  gravity  of  the  part  immersed  will  coincide 
with  the  centre  of  buoyancy. 

668.  The  fluid  and  body  being  homogeneous,  the  centre  of 
gravity  G  {Pig.  223)  will  be  situated  above  the  point  O,  the 
centre  of  buoyancy.  For  let  g  be  the  centre  of  gravity  of 
that  portion  of  the  body  which  lies  without  the  fluid  :  then, 
the  centre  of  gravity  G  of  the  entire  body  will  necessarily 
be  situated  upon  the  line  ^O,  and  between  the  points  g  and  O  ; 
hence,  it  will  be  found  above  the  point  O. 

669.  But  if  the  floating  body  be  heterogeneous,  it  may 
happen  that  the  centre  of  gravity  of  the  entire  body  will  lie 
below  the  centre  of  buoyancy.  For  by  supposing  the  density 
of  the  lower  part  of  the  body  to  be  very  much  greater  than 
that  of  the  upper  portion,  the  centre  of  gravity  of  the  entire 
body  may  be  situated  extremely  near  the  lower  surface  :  but 
the  position  of  the  centre  of  buoyancy  depends  only  on  the 
figure  of  the  part  immersed,  since  the  density  of  the  fluid  is 
supposed  uniform,  and  it  may  therefore  be  situated  at  a 
greater  distance  from  the  lower  surface  of  the  body  than 
the  centre  of  gravity  of  the  entire  mass. 

Hence  we  conclude,  that  the  centre  of  gravity  of  the  float- 
ing body  is  sometimes  situated  above,  and  sometimes  below, 
the  centre  of  buoyancy. 

670.  When  the  body  is  but  partially  immersed,  the  weight 
of  the  immersed  portion  is  less  than  that  of  the  fluid  dis- 
placed, and  the  equilibrium  is  maintained  by  the  weight  of 
that  portion  of  the  body  which  lies  without  the  fluid  :  this 
weight  is  equal  to  the  difiîsrence  of  the  weights  of  the  fluid 
displaced  and  of  the  part  of  the  body  immersed.  If  the  weight 
of  the  body  be  increased,  it  will  sink  to  a  greater  depth,  until 
the  weight  of  the  additional  quantity  of  fluid  displaced  shall 
be  equal  to  the  weight  added. 


368  HYDROSTATICS. 

G71,  Let  us  now  suppose  that  a  body  floating  upon  the 
surface  of  a  fluid  {Fig.  224)  is  deranged  in  a  very  sUght  de- 
gree from  its  position  of  equilibrium,  by  the  application  of 
any  force,  and  let  us  examine  whether  the  body  will  tend  to 
return  to  its  original  position,  or,  on  the  contrary,  to  deviate 
farther  from  it.  Let  ADB  represent  the  immersed  part  of  the 
body  before  derangement,  and  a6D  that  immersed  after  de- 
rangement :  we  suppose  the  new  position  of  the  body  to  be 
such,  that  the  weight  of  the  fluid  displaced  shall  still  be  equal 
to  the  weight  of  the  body,  or  that  ABD  =  a6D.  The  centre 
of  gravity  G  may  be  regarded  as  fixed  during  the  rotation^ 
since  the  forces  will  tend  to  turn  the  system  about  that  point, 
as  though  it  were  immoveable.  The  centre  of  buoyancy 
will  not  retain  its  position  O,  but  will  be  found  nearer  to  the 
portion  CB6,  which,  by  the  rotation,  has  become  immersed  in 
the  fluid  :  and  if  we  suppose,  for  the  sake  of  symplifying  the 
question,  that  the  body  is  divided  symmetrically  by  the  plane 
ABD,  the  centre  of  buoyancy  will  obviously  be  found  in  this 
plane  after  the  derangement.  Let  o  represent  the  centre  of 
buoyancy  in  the  deranged  position,  and  through  o  and  G  let 
perpendiculars  oi  and  Gk  be  demitted  upon  the  line  ab.  If 
an  equilibrium  subsist,  the  weight  of  the  body  and  the  up- 
ward pressure  of  the  fluid  will  be  equal  and  directly  opposed. 

The  first  condition  will  necessarily  be  satisfied,  since  we 
have  supposed  the  volume  of  fluid  displaced  to  remain  un- 
changed; the  second  condition  will  be  fulfilled  when  the 
points  i  and  k  coincide  with  each  other  :  but  if  this  coinci- 
dence should  not  take  place,  the  point  i  may  fall  either  to  the 
right  or  to  the  left  of  the  point  k.  In  the  first  case,  the  pres- 
sure of  the  fluid  applied  at  o  and  acting  upwards,  will  evi- 
dently tend  to  restore  the  body  to  its  primitive  position,,  or  to 
render  the  line  DG  vertical.  But  if  the  point  i  should  fall  to^ 
the  left  of  k,  this  pressure  would  tend  to  turn  the  body  in  a 
contrary  direction  about  the  point  G,  and  would  thus  cause 
it  to  deviate  farther  from  its  original  position. 

If  the  body,  when  deranged  in  a  very  slight  degree  from  its 
position  of  equilibrium,  should  tend  to  resume  its  former  posi- 
tion, the  equilibrium  is  said  to  be  stable  ;  but  if,  on  the  con- 
trary, it  should  tend  to  depart  still  farther  from  this  position, 


EaUILIBRIUM    OF    FLOATING    BODIES.  369 

the  equilibrium  is  called  unstable  ;  when  the  body  neither 
tends  to  return  to  its  original  position,  nor  to  deviate  farther 
from  it,  the  equilibrium  is  said  to  be  one  of  indifference. 

672.  By  examining  the  directions  of  the  pressures  before 
and  after  derangement,  we  shall  find  that  the  lines  OG  and 
oi  perpendicular  to  AB  and  ab  respectively,  are  inclined  to 
each  other,  and  being  contained  in  the  same  plane,  they  will 
intersect  in  some  point  m,  {Fig.  225). 

This  point  is  called  the  metacentre  ;  and  it  appears  from 
Art.  671,  that  when  the  point  G  is  situated  below  w,  the 
extremity  k  of  the  perpendicular  GA;  will  fall  to  the  left  of  the 
point  i,  and  the  equilibrium  will  be  stable  ;  but  if  the  point  G 
be  situated  above  the  point  w,  the  extremity  k  of  the  perpen- 
dicular Ok  will  fall  to  the  right  of  the  point  ?',  and  the  equi- 
librium will  become  unstable.  If  the  points  i  and  k  coincide, 
the  equilibrium  becomes  one  of  indifference. 

673.  Let  it  now  be  required  to  determine  the  position  of 
the  metacentre.  This  point  being  found  upon  the  line  con- 
necting the  centre  of  gravity  and  centre  of  buoyancy  in  the 
primitive  position  of  the  body,  it  will  be  sufficient  to  determine 
its  distance  from  the  point  G,  or  the  point  O. 

For  this  purpose  we  remark,  that  when  the  body  is  slightly 
inclined,  the  line  AB  {Fig.  226)  which  represents  the  profile 
of  the  plane  of  floatation  in  the  primitive  position,  assumes  a 
position  inclined  to  the  new  plane  of  floatation  ab  in  a  certain 
angle  «,  the  portion  ACa  being  at  the  same  time  withdrawn 
from  the  fluid,  and  the  portion  BC6  being  immersed.  Hence, 
the  immersed  portions  of  the  body  in  the  two  positions  will  be, 

aCBD+ACa in  the  primitive  position, 

aCBD  +  BC6  .....  after  the  derangement. 
But  if  y,  g,  and  g'  represent  the  respective  centres  of  gravity 
of  the  volumes  aCBD,  aCA,  and  èCB,  the  centre  of  gravity 
O  of  the  volume  ABRD  will  be  found  by  dividing  the  line  gy 
in  the  inverse  ratio  of  the  volumes  aCBD  and  ACa  ;  and  in 
like  manner,  we  may  find  the  centre  of  gravity  o  of  the 
volume  aèBD  :  thus,  we  shall  obtain  the  proportions 

vol  aCBD  :  vol  aCA  ;  :  O^  :  Oy (400), 

vol  aCBD  :  vol  6CB  ::og'  :oy (401)  ; 

Aa 


370  HYDROSTATICS. 

but  the  second  terms  of  these  proportions  are  equal  to  each 
other  :  for,  the  floating  body  being  supposed  to  displace  the 
same  quantity  of  fluid  after  it  has  been  deranged  as  it  did  in 
its  primitive  position,  the  volumes  ABRD  and  abRD  will  be 
equal  to  each  other  ;  and  if  from  these  equals  we  subtract 
the  common  part  aCED,  there  will  remain  the  volumes  aCA 
and  BC6  equal  to  each  other.  Hence,  we  deduce  from  the 
proportions  (400)  and  (401), 

Og-  :  og'  :  :  Oy  :  oy  ; 
which  proves  that  the  lines  gy  and  g'y  are  cut  proportionally 
by  the  right  line  Oo,  which  line  is  therefore  parallel  to  gg'. 

But,  the  derangement  of  the  body  being,  by  hypothesis, 
extremely  slight,  the  line  gg'  may  be  considered  as  nearly 
coincident  with  the  primitive  plane  of  floatation  ;  and  since 
Oo  is  parallel  to  gg',  this  line  may  be  regarded  as  parallel  to 
the  same  plane, 

674.  To  determine  the  value  of  Oo,  we  deduce,  from  the 
proportion  (400), 

vol  aCBD+vol  aCA  :  vol  aCA  :  :  O^+Oy  :  Oy, 
or, 

vol  ABRD  :  vol  aCA  :  :  gy  :  Oy. 
But  the  similar  triangles  gg'y  and  Ooy  give 

gy.Oy::  gg'  :  Oo  ; 
and  by  comparing  this  proportion  with  the  preceding,  we 

obtain 

vol  ABRD  :  vol  aCA  ::  gg':Oo (402)  ; 

whence, 

vol_aCAx^'  .^„. 

^'-    vol  ABRD     ^^^^^' 

675.  Having  determined  the  value  of  Oo,  we  can  readily 
obtain  that  of  Om  (Fig.  227)  ;  for,  the  lines  Om  and  om 
being  respectively  perpendicular  to  CA  and  Ccr,  the  angles  at 
C  and  m  will  be  equal  ;  and  since  these  angles  are  exceed- 
ingly small,  we  may  regard  the  triangles  ACa  and  Oîno  as 
similar  and  isosceles  :  hence,  we  shall  obtain  the  proportion 

Aa  :  Oo  :  :  Ca  :  viO  ; 
and  therefore, 

^Q^OoxCa 

Aa 


ËQ,lîlLlBRltJM    OP    t^LOATING    BODIES.  371 

6r6i  To  obtain  the  analytical  expressions  for  Oo  and  mO, 
we  remark,  that  the  plane  of  floatation  AB  {Pig.  228),  which 
limits  the  immersed  part  of  the  body  in  its  primitive  position, 
is  replaced  by  the  plane  ab  after  the  derangement  :  these  two 
planes,  being  intersected  by  a  vertical  plane  perpendicular  to 
their  common  intersection,  will  exhibit  the  section  ACa  rep- 
resented in  Fig.  226  ;  and  if  we  continue  to  draw  other  par- 
allel vertical  planes,  we  shall  divide  the  solid  included 
between  the  planes  KAL,  KaL  {Pig:  228)  into  an  infinite 
number  of  elementary  laminae  parallel  to  the  plane  ACa. 

But  it  is  evident,  that  when  the  plane  KAL,  which  in  the 
primitive  position  of  the  body  coincided  with  the  surface  of 
the  fluid,  shall  have  been  detached  from  the  surface,  revolving 
around  the  line  KL,  each  right  line  in  this  plane,  as  CA,  will 
have  described  the  sector  of  a  circle  ;  so  that  the  sections  of 
the  solid  included  between  the  planes  ALB  and  oLb  {Fig. 
228)  by  the  system  of  parallel  vertical  planes,  will  be  repre- 
sented by  the  sectors  ACa,  A'C'a',  A"C"a",  &c.  {Fig.  229). 
But  if  we  assume  the  line  of  intersection  KL  as  the  axis  of 
x,  and  place  the  axis  of  y  in  the  plane  KAL,  the  ordinates  y 
will  be  the  perpendiculars  AC,  A'C,  A"C",  (fcc.  The  infinitely 
small  angle  formed  by  the  planes  KAL  and  KaL  being  every- 
where the  same,  let  the  arc  described  by  a  point  at  the  distance 
unity  from  the  line  KL  be  expressed  by  <y  :  the  arc  described 
by  the  point  A  will  then  be  determined  by  the  proportion 

1 :  41  :  :  AC  or  y  :  arc  Aa  ; 
whence, 

arc  ka=uy (405). 

This  arc  being  multiplied  by  the  half  of  the  radius  y,  we 
shall  obtain  ^ay^^  for  the  area  of  the  sector  ACa  ;  and  this 
area  being  multiplied  by  CC'=dxj  the  portion  of  the  line  KL 
intercepted  between  two  consecutive  sectors,  we  shall  have 
for  the  volume  of  the  solid 

Kaa'A!=\wy^dx, 
which  will  express  the  element  of  the  solid  included  between 
the  planes  KAL  and  KaL.     Hence,  we  shall  have  {Fig.  226) 

vol  A.Ca  =  \é>fy^'dx (406). 

Such  will  be  the  analytical  expression  for  the  second  term  of 
the  proportion  (402). 


372  HYDROSTATICS. 

To  determine  the  value  of  the  third  term,  we  remark  that 
the  Une  C^  {f'ff-  22(3)  being  the  distance  of  the  axis  KL 
{Pig,  229)  from  the  centre  of  gravity  of  the  sohd  KaLA,  we 
shall  determine  this  distance,  by  dividing  the  sum  of  the  mo- 
ments of  the  elementary  solids  by  the  volume  KaLA. 

If  we  consider  the  elementary  sector  ACa  {Fig.  229),  the 
centre  of  gravity  g  of  this  sector  will  be  found  upon  the 
radius  CR=CA  {Fig.  230),  at  a  distance  from  the  point  C 
(Art.  184)  expressed  by 

chordAa 
arc  Aa 
but  the  angle  C  being  supposed  extremely  small,  the  arc  Aa 
may  be  regarded  as  equal  to  the  chord  ;  and  since  CR  is  equal 
to  CA  or  y  {Fig.  228),  the  preceding  expression  will  give  |y 
for  the  distance  of  the  centre  of  gravity  from  the  axis  KL. 
Multiplying  the  elementary  solid  \ay^dx  by  this  distance,  the 
moment  of  this  solid  with  reference  to  this  axis  KL  will 
become  \uy^dx  :  thus,  we  shall  have 
f^uy'^dx=i\\.ç.  sum  of  the  elementary  solids, 
f^iiy^dx=X\ve  sum  of  the  moments  of  the  elementary  solids  : 
and  from  the  property  of  the  moments,  the  distance  Cg  of  the 
centre  of  gravity  of  the  small  solid  CAa  {Fig.  226),  or 
KALa  {Fig.  229),  will  be  expressed  by 

^    f^^^y'dx  ' 
the  quantity  «  being  constant,  this  expression  may  be  re- 
duced to 

^    2jrdx 

^  ^fy'dx 
Ç)77.  The  value  of  C^  will  result  from  the  integrations  here 
indicated  ;  and  that  of  C^'  {Fig.  226)  may  be  obtained  in  a 
similar  manner  ;  but,  if  the  floating  body  be  symmetrical  with 
respect  to  a  vertical  plane  passing  through  the  axis  KL,  as 
will  always  happen  in  the  case  of  a  ship,  we  shall  have 

and  therefore, 


KQtJILlBRIUM    OF    FLOATING    BODIES.  373 

The  volume  of  the  part  immersed,  which  hkewise  enters 
into  the  equation  (403),  can  be  calculated  directly,  when  the 
figure  of  the  vessel  is  supposed  known.  Let  this  volume  be 
denoted  by  V,  and  let  its  value  and  those  of  the  volume  ACa 
and  gg',  given  in  equations  (406)  and  (407),  be  substituted  in 
equation  (403)  :  we  shall  thus  obtain 

and  lastly,  by  substituting  in  equation  (404)  this  value,  and 
that  of  the  arc  Aa,  given  by  equation  (405),  replacing  Ca  by 
y,  we  find 

3V 

Such  is  the  formula  expressive  of  the  distance  of  the  meta- 
cenire  from  the  centre  of  buoyancy. 

678.  When  the  floating  body  is  homogeneous,  and  of  such 
figure  that  its  parallel  sections  will  be  similar,  we  may  rea- 
dily determine  the  position  of  the  metacentre,  without  the 
necessity  of  performing  an  integration.  For  let  a^  represent 
the  area  of  the  section  AEB  (Fig.  231),  which  is  supposed  to 
have  been  determined  by  direct  measurement,  and  let  b  repre- 
sent the  half-breadth  CA  of  this  section  :  the  half-breadths  of 
the  sections  A'E'B',  A"E"B",  «fee.  will  be  represented  by  C'A', 
C'A",  &.C.  or  by  the  ordinates  y  of  the  curve  KAL.  These 
sections  being  by  hypothesis  similar  figures,  they  will  be 
proportional  to  the  squares  of  their  homologous  sides  ;  and 
hence,  we  shall  have 

section  AEB  :  section  A'E'B'  :  :  AC^  :  A'C'^, 
or, 

«2  :  section  A'E'B'  :  :  b^  x  y^\ 
whence, 

section  A'E'B'=^. 
b^ 

The  distance  CC  between  two  consecutive  sections  being 

denoted  by  dx,  we  shall  have 

a^y'^dx 

~~b^ 

for  the  expression  of  the  elementary  solid. 

32 


374 


HYDROSTATICS. 


679.  Let  g  represent  the  centre  of  gravity  of  the  section 
AEB,  which,  in  consequence  of  the  symmetry  of  the  figure, 
will  be  found  on  the  vertical  CE,  The  centre  of  gravity  of 
this  section  having  been  determined,  let  its  distance  from  the 
surface  of  the  fluid  be  denoted  by  n  :  we  shall  then  have, 
from  the  similarity  of  figures, 
,  (  the  distance  of  the  centre  of  ffravitv  of  the  ; 

'  }  section  A'E'B'  from  the  surface  of  the  fluid  )         ' 
whence, 

Multiplying  this  distance  by  the  elementary  solid,  we  shall 
obtain  for  the  moment  of  this  solid,  taken  with  reference  to 
the  surface  of  the  fluid, 

ny     a^y^  , 

and  therefore  the  expression— -/ y ^^f a,-  will  represent   the 

sum  of  the  moments  of  the  elementary  solids  taken  with  refer- 
ence to  the  surface  of  the  fluid.  This  sum  being  equal  to 
the  product  of  the  volume  V  of  the  solid  immersed  by  the 
depth  HG  of  its  centre  of  gravity,  if  this  depth  be  denoted 
by  Gj  we  shall  have 

whence, 

G  =  ; 


-jy^'dx. 


Y.b- 

But  it  has  been  shown  (Art.  677)  that  the  distance  mO  of 
the  metacentre  from  the  centre  of  buoyancy  is  given  by  the 
formula 

mu         ^^      , 
and  if  we  compare  these  two  expressions,  we  shall  find 

G:mO::3na-.2b': 


na^      .  r  ^  J       2  ^^  ly^dx 
or 


EQUILIBRIUM    OF    FLOATING    BODIES.  375 

whence, 

™"=l^ w- 

680.  For  the  purpose  of  applying  this  formula,  let  it  be 
required  to  find  the  metacentre  of  a  rectangular  parallelo- 
piped  ML.  Let  AF  represent  the  intersection  of  the  body- 
by  the  surface  of  the  fluid  {Fig.  232),  supposed  parallel  to 
the  base  NL.  The  depth  AN,  to  which  the  body  must  be  im- 
mersed in  order  that  it  may  be  sustained  in  equilibrio,  will 
depend  on  the  weight  of  the  parallelopiped  and  the  density  of 
the  fluid  (Art.  664)  :  this  depth  may  be  considered  as  deter- 
mined by  experiment  :  the  quantity  a",  which  represents  the 
section  BN,  and  which  will  be  constant  for  all  parallel  sec- 
tions, will  be  determined  immediately  ;  for  we  have 

a^=ABxCE. 
Again,  the  semi-breadth  of  the  section  being  equal  to  |AB, 
there  results 

6  =  iAB=AC; 

and  since  the  centres  of  gravity  of  all  the  sections  are  equally 
distant  from  the  surface  of  the  fluid,  the  centre  of  gravity 
of  the  fluid  displaced  will  be  situated  at  the  same  distance  ; 
so  that  we  shall  have 

W  =  G=:iCE. 

By  substituting  these  values  in  formula  (408),  the  distance 
of  the  metacentre  m,  from  the  centre  of  buoyancy  will  be 
found  equal  to 

,^        SAC^» 
^^=âÂB^E 
or,  by  reduction, 


3CE 

For  example,  if  the  semi-breadth  of  the  parallelopiped  be 
supposed  equal  to  9  feet,  ^ud  the  depth  of  the  part  immersed 
4  feet,  we  shall  find  the  height  of  the  metacentre  above  the 
centre  of  buoyancy  equal  to  6|  feet  ;  if,  therefore,  we  subtract 
from  this  height,  2  feet,  the  depth  of  the  centre  of  buoyancy, 
there  will  remain  4|  feet,  for  the  height  of  the  metacentre 


376  HYDROSTATICS. 

above  the  surface  of  the  fluid.  Hence,  the  centre  of  gravity 
of  the  parallelopiped  should  not  be  more  than  4|  feet  above 
the  surface  of  the  fluid,  if  we  wish  the  equiUbrium  to  be  of 
the  stable  kind. 

681.  As  a  second  example,  let  us  consider  a  vessel  whose 
vertical  sections  below  the  surface  of  the  fluid  are  equal  right- 
angled  isosceles  triangles,  such  as  AEB  {Fig.  233). 

If  the  perpendicular  EC  be  demitted  upon  the  base,  the 
triangle  ABC  will  likewise  be  isosceles,  and  the  height  EC 
will  therefore  be  equal  to  one-half  the  base  AB  :  thus,  the 
quantities  which  enter  into  the  formula  (408)  will  be,  in  the 
present  case, 

a2=area  of  the  triangle  AEB^AC^, 

^=G=iCE, 

h=AC=CE; 
consequently,  by  substituting  these  values  in  formula  (408),  it 
will  reduce  to 

and  if  from  this  value  we  subtract  that  ot  the  distance  of  the 
point  O  below  the  surface  of  the  fluid  which  is  equal  to  ^CE, 
there  will  remain  iCE  for  the  distance  of  the  metacentre 
above  the  surface  of  the  fluid.  Hence,  in  a  prismatic  vessel 
whose  vertical  sections  are  right-angled  isosceles  triangles, 
the  metacentre  will  be  found  at  a  distance  above  the  surface 
of  the  fluid  equal  to  the  distance  of  the  centre  of  buoyancy 
below  the  surface. 

682.  If  we  suppose  the  body  to  be  slightly  deranged  from 
a  position  of  stable  equilibrium,  and  conceive  the  resultant  of 
all  the  upward  pressures  of  the  fluid  to  be  applied  on  its  line 
of  direction,  at  the  metacentre,  we  can  determine  the  circum- 
stances of  oscillation  of  this  body  about  the  centre  of  gravity, 
by  a  method  entirely  analogous  to  that  employed  in  consider- 
ing the  motion  of  the  compound  pendulum.  For  this  pur- 
pose, let  the  origin  of  co-ordinates  be  placed  at  the  centre  of 
gravity,  and  let  the  proper  value  of  y,  be  substituted  in  for- 
mula (337),  which  may  be  put  under  the  form  (338) 

(It     k^'+a'"' 


OSCILLATIONS    OF   FLOATING   BODIES.  377 

This  formula  admits  of  simplification  in  the  present  case, 
from  the  consideration  that  the  oscillations  are  performed 
about  the  centre  of  gravity  ;  and  the  general  expression  of 
the  moment  of  inertia  M.{k^-\-a'')  is  therefore  reduced  to 
Mk^  :  hence,  we  obtain 

^=^ (409). 

dt     k^  ^      ^ 

This  equation,  when  integrated,  will  serve  to  determine  the 
angular  velocity,  and  the  time  of  performing  a  complete  oscil- 
lation. 

683.  To  determine  the  value  of  y,,  which  represents  the 
perpendicular  distance  from  the  axis  passing  through  the  cen- 
tre of  gravity,  about  which  the  oscillations  are  performed,  to 
the  line  of  direction  of  the  upward  pressure,  we  remark,  that 
the  distance  of  the  metacentre  from  the  centre  of  buoyancy 
O  is  expressed  by 

2fy=dx 
3V    * 
Let  this  distance  be  denoted  by  A,  and  the  distance  GO  {Fig. 
234)  by  B  ;  we  shall  then  have 

or,  since  the  point  G  may  fall  above  O,  we  may  likewise  have 

Gm=A— B  ; 
hence  we  may  comprise  the  two  cases  under  the  double  sign, 
by  writing 

Gm=A±B. 

If  the  angle  LmG  {Pig.  234),  formed  by  the  vertical  mh  with 
the  new  direction  of  the  line  GO,  be  represented  by  6,  we  shall 
have  the  relation 

GL=Gm  sinfl; 
or,  replacing  the  sine  by  the  arc,  since  the  arc  is  supposed 
extremely  small,  and  substituting  the  value  of  Gm,  this  equa- 
tion will  become 

GL=(A±B)<»; 
and  by  introducing  this  value  of  y,  in   formula  (409),  we 
shall  obtain 

dcj_g{A±B)ê 
di        ¥^ 


376 


HYDROSTATICS. 


684.  But  the  angular  velocity  a  being  that  which  corres- 
ponds to  the  arc  i  described  with  a  radius  unity,  this  velocity 

will  be  expressed  by  —  ;  and  since  the  arc  «  {Fig.  234)  is  a 

decreasing  function  of  the  time  t^  cU  should  be  affected  with 
the  negative  sign  ;  hence, 

"=^' («o>- 

Multiplying  the  corresponding  terms  of  these  equati(Mis  toge^ 
ther  dt  will  disappear  :  and  there  will  result 

Putting,  for  brevity, 

^^^=E (411), 

and  multiplying  by  2,  we  obtain 

2'Eêd6-^2a>dco=Q. 
Integrating,  we  have 

whence, 

Substituting  this  value  in  equation  (410),  we  obtain 

or,  by  reduction, 

d6 
dt- 


and,  by  integration, 

t=—=  arc  (  cos  =  -'^  )  +C': 
^E        V  ^G/ 

from  which  we  deduce 

'-^=cos[(^-C')v/E]: 

and,  finally. 

,,^C.cos[(^-CVE] 

;        v^ 

685.  When  E  is  negative,  tlie  value  of  o  becomes  imagin- 


SPECIFIC    GRAVITY.  379 

ary,  and  the  oscillatory  motion  cannot  take  place  ;  but  in 
order  that  E  may  be  negative,  the  first  member  of  equation 
(411)  must  likewise  be  negative  ;  and  consequently, 

A  ±  B=a  negative  quantity  : 

this  case  occurs  when  B  exceeds  A,  and  is  affected  with  the 
negative  sign  ;  and  since  A±  B  represents  the  distance  of  the 
centre  of  gravity  from  the  metacentre,  it  follows  that  the  meta- 
centre  will  then  be  situated  below  the  centre  of  gravity,  and- 
the  equilibrium  will  be  unstable.  On  th&  contrary,  if  A±B 
be  positive,  the  metacentre  will  be  found  above  the  centre  of 
gravity,  the  value  of  E  will  be  positive,  and  the  values  of  6 
and  u  will  be  real  :  thus,  the  oscillations  can  be  performed, 
and  the  equilibrium  will  be  of  the  stable  kind. 

686.  The  time  of  oscillation  being  determined  by  a  method 
entirely  similar  to  that  employed  in  investigating  the  circum- 
stances of  motion  of  the  compound  pendulum,  we  may  con- 
clude that  this  time  will  be  independent  of  the  extent  of  the 
arc  through  which  the  oscillations  are  performed,  provided 
the  arcs  be  extremely  small. 

Specific  Gravity — Hydrostatic  Balance — Hydrometer. 

687.  Let  P  represent  the  weight  of  a  body  M  :  if  this  body  be 
immersed  in  a  fluid,  the  buoyant  effort  exerted  by  the  fluid  will 
tend  to  support  the  body,  and  the  force  P'  necessary  to  sustain 
it  will  be  less  than  P,  that  required  previous  to  the  immer- 
sion, by  a  quantity  equal  to  the  weight  of  the  fluid  displaced. 

For  example,  if  M  be  supposed  a  sphere  of  lead  whose 
weight  is  equal  to  eleven  pounds,  and  if  it  be  found  to  weigh 
but  ten  pounds  when  immersed  in  water,  we  should  conclude 
that  the  weight  of  an  equal  volume  of  water  would  be  one 
pound  ;  and  therefore  tfiat  the  weight  of  lead  was  to  that  of 
water  as  eleven  to  one. 

688.  The  specific  gravity  of  any  substance  is  the  ratio 
between  its  weight  and  the  weight  of  an  equal  volume  of 
some  other  substance  assumed  as  the  standard. 

Thus,  in  the  preceding  example,  if  water  be  adopted  as  the 
standard  of  comparison,  the  weight  of  the  sphere  of  lead 


380  HYDROSTATICS. 

being  eleven  times  greater  than  that  of  an  equal  volume  of 
water,  the  specific  gravity  of  lead  will  be  represented  by  the 
number  11. 

The  density  of  a  body  has  been  defined  (Art.  161)  to'be  the 
ratio  between  the  quantity  of  matter  contained  in  the  body  and 
that  contained  in  an  equal  volume  of  some  other  substance 
assumed  as  the  standard  ;  and  since  the  weights  of  bodies  are 
proportional  to  the  quantities  of  matter  which  they  con- 
'iain,  it  follows  that  the  ratio  of  the  weights  of  two  bodies 
will  be  equal  to  the  ratio  of  their  quantities  of  matter. 
Hence,  the  number  expressing  the  specific  gravity  of  a  body 
will  be  the  same  as  that  which  expresses  its  density,  provided 
we  refer  the  density  and  specific  gravity  to  the  same  sub- 
stance as  a  standard. 

In  practice,  it  is  usual  to  adopt  water  as  the  standard  in 
determining  the  specific  gravities  of  solids  and  incompressible 
fluids  ;  and  for  the  purpose  of  rendering  the  comparison  more 
exact,  the  water  is  first  deprived,  by  distillation,  of  any  im- 
purities which  it  may  contain.  The  specific  gravities  of 
gases  and  vapours  are  generally  referred  to  that  of  atmo- 
spheric air. 

689.  The  dimensions  of  all  bodies  being  more  or  less 
aiFected  by  changes  of  temperature,  it  becomes  necessary  to 
adopt  a  standard  temperature,  at  which  experiments  for  the 
determination  of  specific  gravities  may  be  performed.  A 
convenient  temperature  for  this  purpose  is  that  corresponding 
to  60°  of  Fahrenheit's  thermometer,  it  being  easily  obtained 
at  all  times  :  and  the  tables  of  specific  gravities  are  usually 
calculated  for  this  temperature.  When  circumstances  will 
not  permit  the  experiments  to  be  performed  at  the  standard 
temperature,  the  results  obtained  must  be  reduced  to  this  tem- 
perature, by  introducing  a  correction  for  the  change  of  vol- 
ume which  the  substance  would  undergo  if  reduced  to  the 
standard  temperature.  This  correction  is  readily  applied 
when  the  law  of  dilatation  has  been  previously  ascertained. 

690.  If  we  wish  to  determine  the  specific  gravity  of  a  fluid, 
as  olive-oil,  we  may  immerse  successively  the  same  solid  in 
water  and  in  this  fluid  ;  we  shall  thus  be  enabled  to  deter- 
mine the  weights  of  equal  volumes  of  the  two  fluids  ;  and  a 


SPECIFIC    GRAVITY.  381 

comparison  of  these  weights  will  give  the  specific  gravity 

of  the  oil.     For  example,  if  the  sphere  of  lead  weighing 

eleven  pounds  have  its  weight  reduced  to  10.085  lb.  when 

immersed  in  oil,  the  weight  of  the  fluid  displaced  would  be 

equal  to  0.915  lb.  ;  and  since  the  weight  of  an  equal  bulk  of 

0.915 
water  was  found  equal  to  1  lb.,  we  shall  obtain  -^ —  =  0.915, 

for  the  ratio  of  the  weights  of  equal  bulks  of  the  two  fluids  : 
this  number  will  therefore  represent  the  specific  gravity  of  oil. 

From  the  preceding  remarks,  we  may  infer  that  if  two 
bodies  of  unequal  volumes,  suspended  from  the  arms  of  a 
balance,  sustain  each  other  in  vacuo,  the  equilibrii^m  will  not 
be  maintained  when  the  bodies  are  similarly  suspended  in  the 
atmosphere  ;  the  weight  of  the  larger  body  being  most  sup- 
ported by  the  buoyant  eflbrt  of  the  atmosphere. 

691.  The  instrument  usually  employed  for  determining 
with  accuracy  the  specific  gravities  of  bodies,  is  the  hydro- 
static balance.  This  consists  merely  of  a  delicate  balance, 
having  a  small  hook  attached  to  one  of  its  scales,  by  means 
of  which  the  body  can  be  suspended,  for  the  purpose  of  deter- 
mining its  weight  when  immersed  in  a  fluid.  The  body  is 
connected  with  the  hook  by  a  hair  or  slender  thread,  whose 
weight  is  inconsiderable. 

When  we  wish  to  determine  the  specific  gravity  of  a  solid, 
we  place  it  in  the  scale  to  which  the  hook  is  attached,  and  add 
weights  in  the  opposite  scale  until  an  equilibrium  is  produced. 
The  weights  thus  added  will  represent  the  weight  of  the  body 
in  air.  The  body  is  then  attached  to  the  hook  and  im- 
mersed in  water  ;  and  the  weight  necessary  to  be  placed  in 
the  opposite  scale  to  produce  an  equilibrium  will  give  its 
weight  in  wçiter  :  the  difllerence  between  the  weights  in  air 
and  water  will  be  equal  to  the  weight  of  an  equal  volume 
of  water,  and  by  comparing  this  difference  with  the  weight 
in  air,  we  shall  obtain  the  specific  gravity  of  the  substance 
under  consideration. 

This  process  is  slightly  inaccurate  ;  since  the  buoyant 
efforts  exerted  by  the  atmosphere  upon  the  body  when  im- 
mersed in  it,  and  upon  the  weights  introduced  into  the 
opposite  scale,  have  been  neglected.    But  as  the  density  of 


382 


HYDROSTATICS. 


the  atmosphere  is  very  small,  this  omission  will  not  affect  the 
results  materially. 

When  the  given  substance  is  soluble  in  water,  we  deter- 
mine its  specific  gravity  with  reference  to  some  fluid  in  which 
it  is  insoluble,  and  then  compare  the  specific  gravities  of  the 
two  fluids.  If  the  body  be  lighter  than  water,  we  can  con- 
nect it  with  a  heavier  body,  which  will  cause  it  to  sink. 
Then,  having  the  weights  of  the  heavier  and  lighter  bodies, 
and  that  of  the  compound  in  air,  and  having  ascertained  the 
loss  of  weight  sustained  by  the  heavier  body  and  the  com- 
pound when  immersed,  we  can  readily  deduce  the  weight  of 
the  fluid  displaced  by  the  lighter. 

The  specific  gravity  of  a  fluid  may  be  determined  by 
weighing  successively  the  same  body  in  this  fluid  and  in 
water,  and  comparing  the  weights  of  the  equal  volumes 
displaced.  Or  it  may  be  ascertained  by  weighing  the  same 
vessel  when  filled  with  water,  and  with  the  fluid  under  con- 
sideration ;  these  weights,  being  diminished  by  that  of  the 
vessel  when  empty,  will  give  the  relation  between  the  specific 
gravity  of  the  fluid  and  that  of  water. 

692.  The  hydrometer  is  an  instrument  usually  designed  to 
determine  approximatively  the  specific  gravities  of  fluids.  It 
is  composed  of  a  cylinder  of  glass  or  metal,  to  the  lower 
extremity  of  which  a  cup  is  attached  loaded  with  shot  or 
mercury,  and  terminated  at  top  by  a  slender  graduated  wire. 

When  the  hydrometer  is  plunged  into  a  fluid,  the  weight 
with  which  its  lower  extremity  is  loaded  causes  it  to  assume 
a  vertical  position,  and  it  sinks  to  a  greater  or  less  depth, 
according  to  the  specific  gravity  of  the  fluid.  Hence,  that 
division  on  the  graduated  stem  which  corresponds  to  the 
surface  of  the  fluid  will  serve  to  indicate  the  specific  gravity 
of  the  fluid. 

For  example,  if  the  hydrometer  be  immersed  in  distilled 
water  whose  temperature  corresponds  to  60°  Fahrenheit,  the 
surface  of  the  water  will  intersect  the  stem  at  a  certain 
division,  which  we  shall  suppose  to  be  that  marked  10  :  if 
plunged  in  wine,  it  will  sink  deeper,  say  to  the  11th,  12th, 
or  13th  division  ;  and  if  in  brandy,  to  a  still  greater  depth, 


SPECIFIC    GRAVITY.  383 

the  division  indicated  being  dependent  on  the  quantity  of 
alcohol  which  the  brandy  contains. 

The  use  of  this  instrument  evidently  depends  upon  the 
principle,  that  when  a  body  is  immersed  in  a  fluid,  a  portion 
of  its  weight  equal  to  that  of  the  fluid  displaced  will  be  sup- 
ported by  the  buoyant  effort  of  the  fluid  :  thus,  the  heavier 
the  fluid,  the  less  the  depth  to  which  the  hydrometer  will 
sink. 

693.  The  hydrometer,  as  improved  by  Nicholson,  will 
serve  to  determine  the  specific  gravities  of  solids  or  liquids. 
The  instrument  consists  of  a  hollow  copper  ball  A  {Fig.  235), 
to  the  lower  part  of  which  is  attached  a  brass  cup  of  sufficient 
weight  to  maintain  the  hydrometer  in  a  vertical  position 
when  immersed  in  a  fluid.  The  upper  part  of  the  ball 
carries  a  slender  wire  D,  which  supports  a  small  dish  C  des- 
tined to  receive  the  weights.  The  weight  of  the  hydrometer 
is  such  that  the  addition  of  500  grains  in  the  dish  C  will 
just  sink  the  instrument  in  distilled  water,  at  the  temperature 
60°,  luitil  the  svirface  of  the  water  intersects  the  stem  at  its 
middle  point  D.  If,  therefore,  a  body  be  placed  in  the  dish 
C,  and  weights  be  added  until  the  point  D  shall  correspond  to 
the  surface  of  the  water,  the  difference  between  500  grains 
and  the  weights  added  will  express  the  weight  of  the  body. 
The  body  being  then  transferred  to  the  lower  dish  B,  it  will 
be  found  necessary  to  place  additional  weights  in  the  dish  C, 
in  order  to  sink  the  hydrometer  to  the  same  depth:  these 
additional  weights  will  be  equal  to  the  loss  of  weight  sus.- 
tained  by  the  body  when  immersed.  Hence,  the  specific 
gravity  of  the  solid  may  be  readily  determined. 

When  we  wish  to  determine  the  specific  gravity  of  a  fluid 
with  this  hydrometer,  we  immerse  the  instrument  succes- 
sively in  distilled  water  and  in  the  given  fluid,  and  ascertain 
the  weights  necessary  to  be  added  in  each  case  to  the  dish  C, 
in  order  to  sink  it  to  the  same  level.  Then,  the  known 
weight  of  the  instrument  added  to  the  weights  introduced 
into  the  upper  dish  will  give  the  weight  of  the  fluid  dis- 
placed. Thus,  we  can  compare  the  weights  of  equal  volumes 
of  the  two  fluids. 


384  HYDROSTATICS. 


Of  the  Pressure  a?id  Elasticity  of  Atmospheric  Air. 

694.  The  weight  of  the  atmosphere  was  first  recognised  by 
Gahieo.  Torricelh,  his  pupil,  demonstrated  the  existence  of 
this  weight  by  the  following  experiment.  Let  AB  (Fig.  236) 
represent  a  glass  tube,  3  feet  in  length,  filled  with  mercury, 
closed  at  the  lower  extremity  and  open  at  the  upper  :  let  the 
finger  be  applied  to  the  open  extremity,  and  let  the  tube  be 
inverted,  and  its  open  extremity  plunged  in  the  basin  of  mer- 
cury :  on  withdrawing  the  finger,  the  mercury  will  be  found 
to  descend  in  the  tube,  leaving  a  certain  portion  of  it  BE  {Pig. 
237)  unoccupied.  If  the  experiment  be  tried  with  tubes  of 
different  lengths  or  diflîerent  diameters,  the  height  of  the 
column  of  mercury  sustained  in  the  tube  will  be  found,  in 
each  case,  to  be  about  29  or  30  inches  above  the  level  of  the 
fluid  in  the  basin.  This  column  of  mercury  is  sustained  by 
the  pressure  of  the  atmosphere,  arising  from  its  weight; 
which  pressure,  being  exerted  upon  the  surface  CD,  is  suf- 
ficient to  counterbalance  the  weight  of  the  column. 

If  the  experiment  be  performed  with  fluids  of  different 
densities,  the  heights  at  which  they  will  be  supported  will  be 
found  to  differ  :  thus,  if  the  fluid  be  water,  whose  density  is 
to  that  of  mercury  as  1  to  13i,  the  height  of  the  column  will 
be  found  equal  to  30in.  Xl3|=34  feet,  nearly;  the  weight 
of  such  column  being  equal  to  the  weight  of  the  column  of 
mercury. 

695.  The  operation  of  the  common  siphon  is  also  to  be 
referred  to  the  pressure  of  the  atmosphere. 

The  siphon  is  a  bent  tube  having  its  two  branches  of 
unequal  lengths.  The  shorter  branch  EF  {Pig.  238)  being 
plunged  into  the  fluid  contained  in  the  vessel  ABCD,  and  the 
air  being  withdrawn  from  the  siphon,  the  pressure  of  the 
atmosphere  exerted  upon  the  surface  BC  will  cause  the  fluid 
to  rise  in  the  siphon  ;  and  if  the  height  of  the  point  F  be  less 
than  that  at  which  the  atmospheric  pressure  can  sustain  the 
given  fluid,  it  will  pass  into  the  longer  branch,  and  will  be 
delivered  at  the  point  C.  The  current  having  commenced 
in  the  siphon,  it  is  maintained  in  consequence  of  the  superior 


PRESSURE    AND    ELASTIClTV    OF    AIR.  385 

M'-eight  of  the  fluid  in  the  longer  arm  overcoming,  in  part,  the 
pressure  of  the  atmosphere  at  the  point  C,  and  thus  permitting 
the  equal  pressure  of  the  atmosphere  exerted  upon  the  surface 
BC  to  force  the  fluid  up  the  shorter  branch.  Hence,  it  is 
obvious  that  the  point  C  nmst  always  be  below  the  surface 
of  the  fluid  in  the  reservoir  ABCD,  in  order  that  the  siphon 
may  be  effective. 

696.  Air  is  an  elastic  fluid,  which  is  susceptible  of  being 
compressed  into  spaces  which  bear  to  each  other  the  inverse 
ratio  of  the  forces  applied. 

This  may  be  established  experimentally  as  follows  :  Let 
A'BCE  {Fig.  239)  represent  a  curved  tube  closed  at  E  and 
open  at  A'  :  let  mercury  be  introduced  into  the  tube  until  it 
shall  stand  at  the  same  level  CC  in  the  two  branches  :  the 
air  contained  in  the  space  CE  will  then  be  of  the  same  density 
as  the  exterior  air.  If  mercury  be  now  poured  into  the  tube 
until  the  part  ABCD  be  entirely  filled,  the  length  AB  being 
equal  to  30  inches,  the  column  of  air  DE  will  be  found  reduced 
to  one-half  its  original  bulk  CE  :  if  mercury  be  again  intro- 
duced until  it  extend  from  A'  to  d,  the  length  A'h  being  equal 
to  60  inches,  the  volume  of  air  will  be  found  reduced  to  a 
space  Et^  =  iCE. 

This  experiment  establishes  the  law  of  compressibility  ; 
for,  before  the  introduction  of  the  mercury,  the  air  contained 
in  the  space  CE,  being  pressed  by  the  weight  of  the  atmo- 
sphere, must  support  a  pressure  equivalent  to  30  inches  of 
mercury.  When  the  same  volume  of  air  is  caused  to  sustain 
the  additional  pressure  of  a  column  of  mercury  AB=30 
inches,  it  is  reduced  to  one-half  its  original  bulk  ;  and  by  the 
further  addition  of  30  inches,  the  air  is  reduced  to  one-third 
of  this  bulk.  Thus,  it  appears,  that  the  spaces  occupied  by 
the  same  mass  of  air  are  inversely  proportional  to  the  pressures 
applied  ;  and  since  the  densities  of  the  air  are  inveisely  pro- 
portional to  the  spaces  occupied  by  the  same  mass,  it  follows 
that  the  densities  will  be  in  the  direct  ratio  of  the  pressures. 

If  the  mercury  be  withdrawn  from  the  tube,  the  air  will 
expand  and  occupy  the  same  space  as  it  did  previous  to 
compression. 

33 


386  HYDROSTATICS. 


Of  Pumps  for  raising  Water. 

697.  The  pump  is  a  machine  employed  for  the  purpose  of 
raising  water.  There  are  three  principal  kinds  of  pumps, 
viz.  the  sucking  pump,  the  lifting  pump,  and  the  forcing 
pump. 

The  sucking  pump,  represented  in  Fig.  240,  consists  of 
two  tubes  ABDC  and  DCHL,  of  unequal  diameters,  connected 
together  ;  the  first  of  these  is  called  the  sucking  pipe,  and  the 
second  the  body  of  the  pump.  Within  the  body  of  the  pump, 
an  air-tight  piston  MN,  having  a  valve  opening  upwards,  is 
moved  through  the  space  MH,  which  is  called  the  play  of  the 
piston.  At  the  lower  extremity  of  the  body  of  the  pump,  a 
second  valve  ^-,  called  the  sleeping  valve,  is  placed,  which 
likewise  opens  upwards. 

The  lower  extremity  AB  of  the  sucking  pipe  being  im- 
mersed in  a  reservoir  containing  water,  and  the  piston  MN 
being  raised  from  the  position  MN  to  HL,  the  air  contained 
in  the  space  CN  will  expand  and  fill  the  space  CL,  its  density 
and  elastic  force  being  both  diminished  :  at  the  same  time, 
the  air  contained  in  the  pipe  AD,  having  a  density  equal  to 
that  of  the  exterior  air,  will,  in  virtue  of  its  elasticity,  exert 
upon  the  valve  A:,  a  stronger  pressure  than  that  arising  from 
the  elasticity  of  the  rarefied  air  contained  in  the  space  CL  : 
hence,  the  valve  k  will  be  forced  open,  and  the  air  contained 
in  the  interior  of  the  pump  will  acquire  a  density  that  is  uni- 
form throughout,  but  less  than  that  of  the  exterior  air  :  then 
the  pressure  exerted  upon  the  surface  of  the  water  AB  being 
less  than  that  exerted  by  the  atmosphere  upon  the  surface  at 
other  points  of  the  reservoir,  the  water  will  rise  in  the  suck- 
ing pipe  to  the  level  A'B',  such  that  the  weight  of  the  column 
A'B ,  together  with  the  pressure  of  the  rarefied  air  contained 
in  the  pump,  shall  be  equal  to  the  pressure  of  the  exterior 
air.  The  densities  of  the  air  in  the  body  of  the  pump  and  in 
the  sucking  pipe  having  become  equal,  the  valve  k  closes  by 
its  own  weight. 

The  piston  being  then  depressed  from  the  position  HL  to 
MN,  the  air  contained  in  the  space  CL  will  be  compressed 


PUMPS.  387 

into  the  space  CN,  and  its  density  and  elastic  force  will 
become  greater  than  those  of  the  air  contained  in  the  sucking 
pipe  :  the  pressure  on  the  upper  surface  of  the  valve  k  being 
now  greatest,  this  valve  will  continue  closed  during  the  de- 
scent of  the  piston,  and  will  intercept  the  communication  be- 
tween the  sucking  pipe  and  body  of  the  pump  :  hence,  the 
density  of  the  air  in  the  sucking  pipe  will  remain  unchanged, 
and  the  water  will  retain  the  level  A'B'.  When  the  piston 
shall  have  regained  the  position  MN,  it  will  have  compressed 
into  the  space  CN,  not  only  the  quantity  of  air  originally  con- 
tained in  CN,  but  likewise  that  portion  which  was  introduced 
into  the  body  of  the  pump  from  the  sucking  pipe.  The  den- 
sity of  the  air  contained  in  the  space  CN  will  therefore  exceed 
that  of  the  exterior  air,  and  its  elastic  force  will  open  the 
valve  I  :  the  air  contained  in  CN  will  thus  be  restored  to  its 
original  density.  The  piston  being  raised  a  second  time,  the 
air  in  MD  will  be  again  rarefied,  a  portion  of  that  con- 
tained in  A'D  will  pass  into  the  body  of  the  pump,  and  the 
equilibrium  will  be  restored  by  the  water  rising  to  a  new 
level  A"B". 

The  same  operation  being  repeated,  the  water  will  rise 
through  the  valve  k  into  the  body  of  the  pump,  will  pass 
through  the  valve  I  in  the  piston,  and  will  finally  be  delivered 
by  the  spout  Q,R. 

698.  We  will  next  examine  the  mechanism  of  the  lifting 
pump.  In  this  pump,  the  piston  MN  {Fig.  241)  is  situated 
below  the  fixed  valve  /r,  and  being  depressed  from  the  posi- 
tion MN  to  HL,  is  supposed  to  pass  below  the  surface  a'h'  of 
the  water  contained  in  the  reservoir  :  the  piston  contains  a 
valve  opening  upwards,  through  which  the  water  passes, 
regaining  its  level  a'h'.  The  piston  being  then  elevated,  the 
column  of  water  a'L,  which  rests  upon  its  superior  base,  being 
prevented  from  returning  through  the  valve,  will  be  raised 
through  a  height  equal  to  the  play  of  the  piston,  and  will  oc- 
cupy the  space  «N  :  at  the  same  time,  a  vacuum  being  formed 
below  the  piston,  the  water  will  be  compelled  to  follow  the 
piston  in  its  motion  by  the  pressure  of  the  atmosphere  on  the 
surface  of  the  water  in  the  reservoir.  But  the  air  contained 
in  the  space  a'D  being  comprpssed  by  the  elevation  of  tlie 

Bb2 


388  HYDROSTATICS. 

piston,  its  elastic  force  will  become  greater  than  that  of  the 
exterior  air,  and  the  valve  k  will  open,  restoring  the  air  below 
k  to  its  original  density.  The  circumstances  will  then  be 
the  same  as  before  the  first  stroke  of  the  piston,  with  the  ex- 
ception that  a  portion  of  water  has  passed  above  the  piston. 
When  the  piston  is  again  depressed,  the  column  of  water  aN, 
which  rests  upon  it,  will  also  descend,  and  the  air  contained 
in  the  space  Co  will  therefore  be  rarefied.  The  descent  of 
the  water  will  continue  until  the  elastic  force  of  the  rarefied 
air  contained  between  the  valve  k  and  the  surface  of  the 
water,  together  with  the  weight  of  the  column  of  water  raised, 
shall  be  equal  to  the  pressure  of  the  atmosphere  :  the  valve 
in  the  piston  will  then  open,  and  an  additional  quantity  of 
water  will  pass  above  the  piston.  By  repeating  the  process, 
a  certain  portion  of  water  will  pass  above  the  piston  at  each 
stroke  ;  and  reaching  the  valve  k,  will  pass  into  the  body  of 
the  pump,  and  may  be  delivered  at  any  height. 

699.  The  forcing  pump  is  a  combination  of  the  sucking  and 
lifting  pumps.  In  this  pump,  the  piston  MN  {Pig.  242)  is 
without  a  valve,  but  the  lateral  pipe  HE  is  provided  with  one 
at  I,  opening  upwards  ;  and  there  is  a  sleeping  valve  at  L,  as 
in  the  sucking  pump.  The  piston  being  raised,  the  water  rises 
into  the  space  MCDEF,  for  the  reasons  assigned  in  describing 
the  sucking  pump  ;  when  the  piston  is  depressed,  the  water  is 
forced  through  the  valve  I  into  the  tube  HG  ;  and  by  con- 
tinuing the  process,  it  may  be  delivered  at  any  height. 

700.  If  the  dimensions  of  the  sucking  pump  be  improperly 
chosen,  it  may  happen  that  the  water  will  rise  only  to  a  cer- 
tain height.  For  the  purpose  of  discovering  in  what  cases 
this  will  occur,  we  shall  simplify  the  question,  by  supposing 
the  pump  to  be  of  uniform  bore  throughout.  Let  the  water 
be  supposed  to  have  been  raised  to  the  level  ZX  {Pig,  243), 
and  the  piston  to  move  through  the  space  ML  :  call 

a=LN,  the  play  of  the  piston, 

6= LB,  the  height  of  the  piston  at  its  greatest  elevation 

above  the  surface  of  the  water  contained  in  the 

reservoir, 
.T=the  distance  LX. 
When  the  piston  is  raised  from  the  position  MN  to  HL,  the 


PUMPS.  389 

air  which  was  previously  contained  in  the  space  ZN  will 
occupy  the  space  ZL,  and  its  elasticity  will  therefore  be 
diminished  in  the  ratio  of  LX  to  NX  ;  so  that  if  R  represent 
the  elastic  force  of  the  air  contained  in  the  space  NZ,  and  R' 
the  elastic  force  of  the  rarefied  air  contained  in  LZ,  we  shall 
have 

LX  :  NX  :  :  R  :  R'  ; 
or, 

X  :  X — a  ;  :  R  :  R'  : 
whence, 

,r — a 


R'=R: 


X 

But  the  air  contained  iw.  the  space  NZ  being  of  the  same 

density  with  the  exterior  air,  its  elastic  force  will  be  properly 

measured  by  the  weight  of  a  column  of  water  whose  base  c  is 

equal  to  the  surface  MN,  and  whose  height  is  equal  to  34  feet. 

Let  this  height  be  denoted  by  h  ;  the  density  of  water  being 

supposed  equal  to  unity,  and  the  force  of  gravity  being  denoted 

by  g^  we  shall  have 

R=cA^. 

This  value,  substituted  in  the  preceding  equation,  gives 

T,,    x—a  , 
R'= dig. 

But  it  is  evident  that  when  an  equilibrium  subsists,  the  elastic 
force  of  this  rarefied  air,  together  with  the  weight  of  the 
column  of  water  BZ,  must  be  just  sufficient  to  counterbalance 
the  pressure  of  the  atmosphere,  which  tends  to  produce  the 
ascent  of  the  water.  The  weight  of  the  column  of  water 
ABXZ  will  be  expressed  by  ^c  X  BX,  or  gc  X  ^—x)  ;  and  the 
pressure  exerted  by  the  atmosphere  will  be  expressed  by  the 
column  gch  ;  hence,  we  shall  have,  in  case  of  an  equilibrium, 

~  gch-\-{h—x)gc=gchy 


X 

or,  by  suppressing  the  common  factor  gc^ 
X — a 


-h  +  b — x=h. 


But,  if  it  were  required  that  the  water  should  rise  above  the 
level  ZX,  it  would  then  be  necessary  that  the  atmospheric 


390 


HYDROSTATICS. 


pressure  should  exceed  that  arising  from  the  weight  of  the 
column  ZB,  and  the  elastic  force  of  the  air  contained  in  the 
space  ZL  :  we  shall  consequently  have 

-h-\-h — x<h. 


X 

Let  z  represent  the  excess  of  the  second  member  of  this  in- 
equality ;  then 

-h-\-h — x-\-z=h  : 

X 

or,  by  reduction, 

— ah-\-hx—x^-\-zx=^  : 
whence^ 

If  we  make  2;=0,  the  water  will  cease  to  rise,  and  we  shall 
then  have 

These  two  values  of  x  will  be  real  so  long  as  —  exceeds  ah  : 

if,  therefore,  this  condition  be  fulfilled,  there  will  be  two  points 
at  which  the  water  will  stop  :  but  if,  on  the  contrary,  ah 

should  exceed  —,  the  values  of  x  will  become  imaginary,  and 

there  can  be  no  point  at  which  the  water  will  cease  to  rise. 
Such  is  the  condition  requisite  to  ensure  the  effective  per- 
formance of  the  sucking  pump. 

701.  With  the  lifting  pump,  the  water  can  be  raised  to  any 
lieight,  provided  sufficient  force  be  applied  to  the  piston. 
For,  let  the  water  be  supposed  to  have  risen  to  the  level  EF 
{Fig.  241),  the  water  in  the  reservoir  standing  at  the  level  aè, 
above  the  piston.  Then,  the  column  included  between  the 
surfaces  ah  and  MN  being  supported  by  the  pressure  of  the 
contiguous  fluid,  the  piston  MNP  will  be  loaded  only  with 
the  weight  of  the  column  extending  from  ah  to  EF. 

702.  But  if  the  level  of  the  fluid  in  the  reservoir  be  sup- 
posed at  a'6'  below  the  piston,  the  weight  P  of  the  column  of 
water  included  between  MN  and  a'6'  must  be  supported  by 


AIR-PUMP.  391 

the  pressure  of  the  atmosphere  exerted  upon  the  surface  of 
the  water  in  the  reservoir.  Hence,  the  pressure  of  the  atmo- 
sphere exerted  upon  the  upper  base  of  the  piston,  through  the 
column  EN,  will  exceed  that  which  is  exerted  upon  the  lower 
base  through  a'N,  by  the  weight  of  the  column  a'N  ;  for  this 
weight  counteracts  in  part  the  pressure  exerted  by  the  atmo- 
sphere upon  the  water  in  the  reservoir.  Thus,  the  piston  MN 
being  urged  downwai'ds  by  the  weight  of  the  column  MF 
situated  above  it,  and  hkewise  by  the  difference  of  the  atmo- 
spheric pressures,  which  is  equal  to  the  weight  of  the  column 
a'N,  the  effect  will  be  the  same  as  though  the  piston  sup- 
ported a  column  of  water  whose  base  is  MN,  and  whose 
altitude  is  equal  to  the  distance  between  the  levels  a'h' 
and  EF. 

It  thus  appears,  that  with  a  sufficient  effort,  the  water  may 
be  raised  to  any  height  by  the  lifting  pump,  the  fixed  valve 
k  being  supposed  near  the  surface  of  the  water. 

The  same  principles  will  serve  to  estimate  the  force  neces- 
.sary  to  raise  the  water  in  the  sucking  pump. 

Of  the  Air-pump. 

703.  In  examining  the  properties  of  various  substances,  it 
is  frequently  necessary  to  withdraw  them  from  the  action  of 
the  atmosphere,  and  it  therefore  becomes  desirable  that  we 
should  have  it  in  our  power  to  exhaust  the  air  from  a  vessel 
in  which  the  substance  has  been  deposited.  This  vessel  is 
called  the  receiver,  and  is  usually  constructed  of  a  transparent 
substance,  such  as  glass,  in  order  that  we  may  have  an  opportu- 
nity of  observing  the  effects  produced  on  the  substance  under 
consideration  by  the  withdrawal  of  the  atmospheric  air. 

704.  The  machine  employed  to  exhaust  the  air  is  called 
an  air-pump,  and  the  term  vacuimi  is  applied  to  the  space 
from  which  the  air  has  been  extracted. 

705.  The  general  principles  upon  which  the  operation  of 
this  machine  depends,  will  be  readily  understood  by  a  refer- 
ence to  Fig.  244.  A  represents  a  section  of  the  glass  receiver 
which  rests  upon  the  plate  BC,  the  lower  edge  of  the  receiver 
and  the  plate  being  ground  exactly  plane,  so  that  their  con- 


392  HYDROSTATICS. 

tact  may  be  as  perfect  as  possible.  The  edge  of  the  receiver 
being  previously  smeared  with  a  little  sweet  oil,  the  air  will 
be  effectually  prevented  from  penetrating  between  the 
receiver  and  plate. 

The  plate  BC  is  perforated  by  a  cavity  DE,  which  commu- 
nicates with  the  cylindrical  barrel  CF,  in  which  an  air-tight 
piston,  having  a  valve  opening  upwards,  is  worked  by  means 
of  a  handle  H.  At  the  bottom  of  the  barrel  is  placed  a  second 
valve  E,  likewise  opening  upwards. 

706.  Let  it  now  be  supposed  that  the  piston  has  been 
depressed  until  it  has  reached  the  valve  E  ;  the  air  in  the 
receiver,  barrel,  and  communicating  pipe  being  of  the  same 
density  as  the  exterior  air,  and  the  valves  being  closed  by 
their  own  weight.  Then,  if  a  force  be  applied  to  raise  the 
piston,  the  valve  P  will  remain  closed,  and  a  vacuum  would 
be  left  between  the  piston  and  the  valve  E,  provided  the 
weight  of  the  valve  E  were  sufficient  to  overcome  the  pres- 
sure exerted  upon  its  under  surface  by  the  elastic  force  of 
the  air  contained  in  the  receiver  and  communicating  pipe  : 
this,  however,  not  being  the  case,  the  valve  E  will  be  forced 
open,  and  a  portion  of  the  air  contained  in  the  pipe  and 
receiver  will  pass  into  the  barrel,  until  the  density  of  the  air 
becomes  uniform  throughout.  This  effect  will  continue  until 
the  piston  has  reached  its  highest  position,  and  the  valve  E 
will  then  close  by  its  own  weight.  The  piston  being  then 
depressed,  the  valve  E  will  remain  closed,  and  the  air  con- 
tained in  the  barrel  being  compressed  into  a  smaller  space,  its 
elastic  force  will  be  increased,  will  become  greater  than  that 
of  the  exterior  air,  and  will  finally  overcome  the  weight  of 
the  valve  P,  causing  it  to  open,  and  thus  reducing  the  density 
of  the  air  contained  in  the  barrel  to  an  equality  with  that  of 
the  exterior  air  :  this  effect  will  only  cease  when  the  piston 
has  been  forced  to  the  bottom  of  the  barrel. 

It  thus  appears  that  by  a  single  ascent  and  descent  of  the. 
piston,  a  portion  of  air  has  been  withdrawn  from  the  receiver 
and  pipe  of  communication.  The  portion  withdrawn  will 
obviously  bear  the  same  ratio  to  the  quantity  originally  con- 
tained in  the  receiver  and  pipe  that  the  capacity  of  the  barrel 
bears  to  the  sum  of  the  capacities  of  the  barrel,  pipe,  and 


AIR-PUMP.  393 

receiver.  By  a  repetition  of  the  same  process,  a  second 
quantity  can  be  withdrawn,  and  the  operation  may  be  con- 
tinued until  the  exhaustion  has  been  carried  to  the  desired 
extent. 

707.  Since  the  quantity  of  air  withdrawn  at  each  ascent 
and  descent  of  the  piston  forms  but  a  part  of  that  previously 
contained  in  the  receiver  and  pipe,  it  is  obvious  that  a  perfect 
vacuum  can  never  be  produced  by  the  operation  of  the  pump. 
The  weight  of  the  lower  valve  likewise  opposes  an  obstacle 
to  the  entire  exhaustion  ;  for,  whenever  the  air  contained  in 
the  receiver  and  pipe  shall  have  had  its  elastic  force  so  far 
reduced  as  to  be  incapable  of  raising  the  valve  E,  the  pump 
will  necessarily  cease  to  exhaust.  This  difficulty  may,  how- 
ever, be  obviated,  by  causing  the  valve  E  to  open  by  means 
of  a  mechanical  connexion  with  the  piston. 

708.  As  it  is  frequently  necessary  to  produce  a  very  high 
degree  of  exhaustion,  it  becomes  interesting  to  ascertain  the 
density  of  the  air  remaining  in  the  receiver  after  any  given 
number  of  strokes  of  the  piston  ;  and  since  the  portion  with- 
drawn at  each  double  stroke  bears  a  constant  relation  to  that 
remaining,  this  density  may  be  readily  estimated.  Thus,  if 
we  denote  by  6,  />,  and  r  the  respective  capacities  of  the 
barrel,  pipe,  and  receiver,  and  by  d  the  original  density  of  the 
air,  we  shall  have  the  proportion 

h-\-v-{-i^  :  p-^-r  ::  d'.  d-^ =density  after  the  first  double 

o-f-^-fr  ' 

stroke. 

In  like  manner, 

b-{-v-\-r:'p-\-r::d~ :  di-^ p=densitv  after  the 

^  ^  b+p+r       \b+p  +  r/  ^ 

second  double  stroke. 
And  generally, 
,+p+r  :f+r  :  :  d  (j|±^,)""'  :  d  (j^J'  =  density 

after  the  nih  double  stroke. 

For  the  purpose  of  illustrating  the  rate  of  exhaustion,  we  will 
suppose  that  the  capacity  of  the  barrel  is  one-fourth  of  the  sum 
of  the  capacities  of  the  receiver  and  pipe  ;  then,  we  shall  have 


394  HYDROSTATICS. 

fc=i(p+r)=i(6+;'+r); 
and  the  density  after  the  first  double  stroke  will  be 

d-^ =d—=^d. 

b-\-p+r      5b     ' 

Thus,  by  the  first  double  stroke  of  the  piston,  one-fifth  of  the 
air  contained  in  the  receiver  and  pipe  will  be  withdrawn,  and 
the  quantity  remaining  will  be  four-fifths  of  the  original 
quantity.  The  density  after  the  second  stroke  will,  in  like 
manner,  be  four-fifths  of  that  after  the  first,  or  ||  of  the  ori- 
ginal density  ;  and  after  the  third,  the  density  will  be  reduced 
to  -fYjj  or  nearly  one-half  It  thus  appears  that  every  three 
strokes  will  reduce  the  density  nearly  one-half;  and  conse- 
quently, that  after  twenty-seven  strokes,  the  air  would  be 
reduced  to  about  one-five-hundredth  of  its  original  density. 

709.  The  preceding  calculation  is  based  upon  the  suppo- 
sition that  the  relative  capacities  of  the  barrel,  pipe,  and 
receiver  have  been  accurately  ascertained,  and  that  the 
mechanical  construction  of  the  pump  is  perfect,  neither  of 
which  conditions  is  strictly  fulfilled  :  and  as  it  is  frequently 
necessary  to  know  the  precise  degree  of  exhaustion  that  has 
been  attained,  it  becomes  important  to  have  a  gauge,  or  index, 
by  the  aid  of  which  we  may  ascertain  the  density  of  the 
remaining  air  at  any  moment.  The  instruments  commonly 
employed  for  this  purpose  are, 

1°.  The  barometer  gauge,  which  consists  of  a  straight 
glass  tube  about  thirty-two  inches  in  length,  and  open  at 
both  extremities.  The  tube  is  placed  in  a  vertical  position, 
its  upper  extremity  communicating  with  the  receiver  of  the 
pump,  and  its  lower  being  immersed  in  a  basin  of  mercury. 
When  the  process  of  exhaustion  has  been  commenced,  the 
air  in  the  tube  being  rarefied,  the  pressure  of  the  atmosphere 
upon  the  surface  of  the  mercury  in  the  basin  will  cause  the 
mercury  to  rise  in  the  tube,  and  the  height  at  which  it  stands 
will  indicate  the  difference  between  the  exterior  and  interior 
pressures.  These  pressures  are  in  the  direct  ratio  of  the 
densities  of  the  air.  The  principal  inconvenience  of  this 
gauge  arises  from  the  necessity  of  having  a  barometer  with 
which  to  ascertain  the  pressure  of  the  exterior  air  at  the  same 
time. 


AIR-PUMP.  395 

2".  The  short  barometer  gauge  is  formed  of  a  tube  eight 
or  ten  inches  in  length,  open  at  one  extremity,  and  filled  with 
mercury.  This  tube  being  inverted,  and  immersed  at  its 
open  extremity  in  a  basin  of  mercury,  the  pressure  of  the 
atmosphere  upon  the  surface  of  the  mercury  in  the  basin  will 
retain  the  tube  entirely  full.  This  apparatus  being  placed 
under  a  receiver  which  communicates  with  that  of  the  pump, 
and  the  rarefaction  being  commenced,  the  short  tube  will 
remain  full  until  the  density  of  the  air  in  the  receiver  has 
been  so  far  reduced  that  its  elastic  force  is  insufficient  to  sup- 
port a  column  of  mercury  of  a  length  equal  to  that  of  the  tube. 
The  mercury  in  the  tube  will  then  fall,  and  its  height  at 
any  moment  will  indicate  the  pressure  of  the  air  within. 
This  gauge  is  evidently  unfit  for  use  when  only  a  moderate 
degree  of  exhaustion  is  required. 

3°.  The  siphon  gauge  is  composed  of  a  short  bent  tube, 
having  two  parallel  branches,  one  of  which  is  closed,  and  the 
other  open.  The  closed  branch  being  filled  with  mercury, 
and  the  tube  being  placed  with  the  bend  downwards,  the 
mercury  will  be  supported  in  that  branch  by  the  pressure  of 
the  exterior  air.  The  tube  is  then  placed  beneath  a  receiver, 
and  acts  upon  the  same  principle  as  the  short  barometer 
gauge,  the  bend  in  the  tube  serving  as  a  substitute  for  the 
basin  of  mercury.  This,  also,  is  only  applicable  when  a  con- 
siderable degree  of  rarefaction  is  required. 

710.  The  working  of  the  piston  being  opposed  by  the 
pressure  of  the  atmosphere  on  its  superior  surface,  and  this 
difficulty  constantly  increasing  as  the  rarefaction  proceeds,  it 
has  been  found  advantageous  to  adapt  a  second  barrel  to  the 
pump,  whose  piston  shall  descend  whilst  that  of  the  first 
barrel  ascends, — and  the  reverse.  The  rods  of  the  pistons 
have  the  form  of  a  rack  whose  teeth  engage  in  those  of  a 
wheel  which  is  turned  by  a  winch.  The  pressures  on  the 
pistons  are  thus  caused  to  oppose  each  other,  and  the  pump 
works  with  much  greater  ease.  The  rapidity  of  the  exhaus- 
tion is  likewise  doubled  by  this  arrangement. 

711.  If  the  construction  of  the  pump  be  such  as  to  require 
the  lower  valves  to  be  opened  by  the  elasticity  of  the  air 
remaining  in  the  receiver,  the  operation  of  the  pump  will  evi- 


396  HYDROSTATICS. 

dently  cease  whenever  the  rarefaction  has  been  carried  so  far 
that  the  weight  of  the  lower  valve  is  sufficient  to  overcome 
the  elastic  force  of  the  air  within.  To  obviate  this  inconve- 
nience, the-  lower  valves  are  opened  and  closed  by  the  motions 
of  the  piston,  as  shown  in  Fig.  245,  which  represents  a  sec- 
tional view  of  one  of  the  most  approved  pumps.  The  dis- 
position of  the  several  parts  has  been  somewhat  altered,  for 
the  purpose  of  exhibiting  them  more  clearly. 

A  represents  the  glass  receiver  resting  upon  the  ground 
glass  plate  BC,  and  communicating  by  the  cavity  DFG  with 
the  tvvo  pump  barrels  VR  and  V'R'.  The  receiver  likewise 
communicates  by  the  cavity  svy  with  the  barometer  gauge 
yz,  immersed  in  the  vessel  of  mercury  M,  and  with  the  siphon 
gauge  vx.  E  is  a  stopcock  for  cutting  off  the  communication 
between  the  receiver  and  the  barrels  when  the  exhaustion  has 
been  effected,  and  E'  a  second  stopcock  for  re-admitting  the 
external  air.  In  the  best  pumps,  the  barrels  are  made  of 
glass,  to  prevent  the  corrosion  which  would  take  place  by  the 
action  of  the  oil  with  which  the  pistons  are  lubricated  to 
render  them  air-tight  :  for  similar  reasons,  the  pistons  are 
sometimes  made  of  steel.  The  racks  L  and  L'  of  the  pistons 
are  worked  by  the  wheel  W,  which  is  turned  alternately  to 
the  right  and  left  by  the  winch  H.  The  lower  valves  Y  and 
V  are  metallic,  and  have  the  form  of  a  conic  frustrum.  To 
the  back  of  the  valve  is  attached  a  slender  rod  VR,  which 
passes  through  an  air-tight  hole  in  the  piston  P,  and  carries 
near  its  upper  extremity  a  small  projection  or  shoulder. 
When  the  piston  is  raised,  the  friction  of  the  valve-rod  which 
passes  through  it  causes  the  rod  likewise  to  rise,  opening  the 
lower  valve  V:  but  this  upward  motion  is  soon  checked  by 
the  shoulder  coming  into  contact  with  the  top  of  the  barrel, 
and  the  rod  then  slides  through  the  hole  in  the  piston. 
Again,  when  the  piston  is  depressed,  it  carries  with  it  the 
valve-rod  RV,  closing  the  valve  at  the  bottom  of  the  pump, 
and  the  descent  of  the  piston  is  then  continued  by  sliding 
along  the  rod. 

712.  The  valves  of  the  pistons  are  variously  constructed. 
In  some  instances  they  are  metallic,  resting  upon  a  metallic 
bed  ;  and  in  others,  they  are  composed  of  strips  of  oiled  silk, 


AIR-PUMP.  397 

bladder,  or  parchment,  stretched  across  an  opening  in  the  pis- 
ton, and  ahernately  allowing  and  preventing  the  communica- 
tion between  the  air  beneath  the  piston  and  the  exterior  air. 
During  the  ascent  of  the  piston,  the  valve  remains  closed  by 
the  stronger  pressure  of  the  atmosphere  on  its  upper  surface, 
and  when  the  piston  descends,  the  compressed  air  beneath  it 
will  force  open  the  valve.  This  latter  condition  will  always  be 
fulfilled,  whatever  may  be  the  degree  of  exhaustion,  provided 
the  piston  can  be  forced  into  actual  contact  with  the  bottom 
of  the  barrel. 

713,  The  pistons  are  usually  composed  of  two  metallic 
plates,  which  carry  between  them  a  packing  of  leather  soaked 
in  oil.  The  distance  between  these  plates  can  be  varied  by 
means  of  a  powerful  screw  ;  and  by  the  application  of  a  proper 
degree  of  pressure,  the  packing  is  caused  to  fit  the  barrel 
with  accuracy. 

714.  By  the  aid  of  the  air-pump  we  are  enabled  to  exhibit 
many  of  the  most  important  properties  of  atmospheric  air  : 

1°.  The  weight  of  the  air  may  be  shown  by  screwing  a 
vessel  provided  with  a  stopcock  to  the  air-pump,  and  ex- 
hausting the  air  from  within  it.  The  weight  of  the  vessel 
will  be  diminished  by  about  ^\  of  a  grain  for  every  cubic 
inch  of  air  that  has  been  withdrawn. 

2^.  The  pressure  of  the  atmoi^nhere  is  rendered  evident  by 
the  difficulty  with  which  the  receiver  is  removed  from  the 
plate  of  the  pump  after  the  air  within  it  has  been  withdrawn. 

A  small  strip  of  bladder  being  stretched  across  the  moutJj 
of  an  open  receiver,  and  the  air  exhausted  from  beneath,  the 
bladder  will  be  ruptured  by  the  pressure  of  the  exterior  air. 

Two  brass  hemispheres,  being  ground  so  as  to  fit  accurately 
to  each  other,  and  attached  to  the  pump,  cannot  be  separated 
without  great  difficulty  after  the  air  has  been  exhausted  from 
the  space  enclosed  by  them.  The  pressu  e  of  the  atmo- 
sphere is  found  to  be  equivalent  to  about  15  lb.  for  each 
square  inch  of  surface  exposed  to  its  action. 

3°.  The  elasticity  of  the  air  may  likewise  be  shown  by 
various  experiments.  If,  for  example,  a  bladder  containino-a 
small  quantity  of  air  be  enclosed  in  a  receiver,  from  which 
the  air  can  be  extracted,  the  elasticity  of  the  air  contained  iii 

34 


398  HYDROSTATICS. 

the  bladder  will  cause  it  to  distend  when  the  exterior  pres- 
sure is  removed  ;  and  on  the  re-admission  of  the  air  into  the 
receiver,  the  bladder  will  again  collapse. 

If  a  light  glass  bulb,  having  an  opening  in  its  lower  surface, 
b«  loaded  with  weights  so  that  it  will  just  sink  in  a  vessel  of 
water  when  the  bulb  is  partially  filled  with  water  ;  upon 
withdrawing  the  air  from  the  receiver  in  which  the  vessel  of 
water  has  been  deposited,  the  portion  of  air  contained  in  the 
bulb  will  expand,  expelling  a  portion  of  the  water  through 
the  orifice  in  the  bottom  of  the  bulb.  The  bulb  and  weight 
will  thus  be  rendered  specifically  lighter  than  water,  and  will 
consequently  rise  to  the  surface  of  the  fluid  in  the  vessel  : 
upon  re-admitting  the  air  into  the  receiver,  a  portion  of  water 
will  be  forced  into  the  bulb,  and  it  will  again  sink. 

4°.  The  resistance  of  the  air  to  the  motion  of  bodies  may 
be  exhibited  by  allowing  two  bodies  of  very  unequal  den- 
sities to  fall  in  the  exhausted  receiver  of  the  air-pump,  and 
in  the  same  receiver  after  the  re-admission  of  the  air.  When 
the  bodies  fall  in  vacuo,  they  will  reach  the  bottom  of  the 
receiver  at  the  same  instant  ;  but  when  the  receiver  contains 
air,  the  denser  body  being  least  retarded  by  the  resistance 
which  the  air  offers,  it  will  fall  through  the  height  of  the 
receiver  in  much  less  time  than  that  required  by  the  rarer 
body. 

Many  other  experiments  may  be  contrived  to  illustrate  the 
properties  of  air,  but  it  is  unnecessary  to  notice  them  in  this 
place. 

Of  the  Barometer. 

715.  The  barometer  is  composed  essentially  of  a  bent  tube 
ABC  [Fig.  246),  closed  at  A,  and  open  at  C,  and  filled  with 
mercury  throughout  the  portion  NMBEF.  The  air  is  sup- 
posed to  have  been  exhausted  from  the  space  AMN,  and  the 
column  of  mercury  included  between  the  planes  MN  and 
DFE  is  supported  by  the  pressure  of  the  atmosphere  upon 
the  surface  FE.  This  column  is  usually  about  thirty  inches 
in  length,  when  the  barometer  is  placed  at  the  level  of  the 
ocean. 


BAROMETER. 

716.  This  instrument  serves  to  indicate  the  changes  which 
are  constantly  taking  place  in  the  pressure  of  the  atmosphere  ; 
for,  when  the  pressure  becomes  greater,  the  length  of  the 
column  of  mercury  which  it  can  sustain  is  necessarily 
increased,  and  the  mercury  therefore  rises  in  the  tube  AD  : 
but  if,  on  the  contrary,  the  pressure  of  the  air  should  dimin- 
ish, the  length  of  the  column  will  undergo  a  corresponding 
diminution. 

The  pressure  of  the  atmosphere  at  any  point  being  that 
due  to  the  weight  of  a  column  of  air  extending  from  that 
point  to  the  top  of  the  atmosphere,  it  follows  that  this 
pressure  will  decrease  as  we  ascend  above  the  earth's  surface, 
and  consequently,  that  the  height  of  the  mercurial  column 
will  diminish. 

717.  This  principle  has  been  employed  to  determine  the 
difference  of  level  of  two  places  situated  at  unequal  distances 
above  the  surface  of  the  earth.  For  the  purpose  of  investi- 
gating a  formula  which  shall  be  applicable  to  this  object,  we 
shall  denote  by 

//.' the  height  of  the  mercurial  column  at  the  lower  station, 

h the  height  of  the  mercurial  column  at  the  upper  station, 

D'  and  D  the  corresponding  densities  of  the  atmosphere  at 

the  two  stations. 
Then,  if  we  suppose  the  axis  of  z  to  be  vertical,  the  general 
equation  of  equilibrium  of  heavy  fluids  as  obtained  in  Art. 

655,  will  be 

d2i=T)gdz* 
Let  the  origin  be  assumed  at  the  lower  station,  and  let  the 
co-ordinates  z  be  reckoned  positive  upwards  ;  then,  as  we 
ascend  in   the  atmosphere,  the   pressure  arising   from   the 
weight  of  the  superincumbent  strata  will  diminish,  and  the 

*  This  result  may  be  obtained  directly  by  considering  a  column  of  the  atmo- 
sphere, whose  base  AB  {Fig.  247)  is  the  unit  of  surface  :  the  pressure  sus- 
tained by  this  base  is  measured  by  the  \\'eight  of  the  column  of  air  ABDC 
extending  to  the  top  of  the  atmosphere  ;  and  the  elementary  pressure  dp  will  be 
represented  by  the  weight  of  a  column  having  the  same  base,  and  a  height  equal 
to  dz.  The  base  of  this  elementary  column  being  equal  to  unity,  its  volume 
will  be  expressed  by  1  Xdz,  or  dz,  and  its  mass  by  Ddz  :  thus,  gDdz  represents 
the  weight  which  will  measure  the  elementary  pressure  dp.  This  result  will 
obviously  be  independent  of  the  particular  form  given  to  the  base  AB  which 
has  been  assumed  as  the  superficial  unit. 


400  HYDROSTATICS. 

density  of  the  air  will  undergo  a  corresponding  decrease. 
Thus,  the  pressure  p  being  a  decreasing  function  of  the  alti- 
tude z,  dp  and  dz  will  be  affected  with  contrary  signs  : 
hence,  the  preceding  equation  should  be  written 

dp=—Dgdz (412). 

If  the  difference  of  level  of  the  two  places  be  but  slight,  the 
force  of  gravity  g-  may  be  regarded  as  constant  :  and  hence 
we  shall  obtain,  by  integration, 

^—IM («^>- 

But  it  has  been  shown  (Arts.  651  and  696)  that  when  the 
temperature  is  supposed  constant,  the  pressure  and  density 
are  proportional  to  each  other  ;  hence,  if  P  denote  the  pressure 
capable  of  producing  a  density  represented  by  unity,  we  shall 
have 

p=PD; 
and  therefore, 

dp=VdT): 

this  value  substituted  in  equation  (41 3)  gives 
__P    /*dD 
^-     gJ    D-' 
and  by  effecting  the  integration  indicated,  there  results 

g- 
To  determine  the  constant,  we  remark,  that  \vhen  z=0,  the 
density  becomes  that  which  we  have  supposed  to  exist  at  the 
lower  station,  and  which  has  been  denoted  by  D'.     Thus,  the 
preceding  equation  becomes 

0=-?logD'+C; 
g- 
eliminating  C  between  this  equation  and  the  preceding,  we 
find 

z=^  (log  D'-log  D), 


or. 


p,    jy 


But  the  densities  being  proportional  to  the  pressures,  they 


BAROMETER.  401 

will  likewise  be  proportional  to  the  observed  altitudes  of  the 
mercurial  column  :  hence, 

h:  h'  ::D  :  D',  or^=5.'; 
h      U 

D' 
this  value  of  --  being  substituted  in  that  of  z,  we  obtam 

P,      h' 

g     ^A 

h' 
The  logarithm  of  —,  which  appears  in  this  expression,  apper- 
tains to  the  Naperian  system  :  if  therefore,  we  represent  by 

h' 
Log—,  the  tabular  logarithm 
h 

the  modulus,  we  shall  have 
and,  by  substitution. 


h'  h! 

Log—,  the  tabular  logarithm  of —,  and  by  M  the  reciprocal  of 
h  h 


MLog|'=log^; 


MP         h! 

g  h 

718.  To  determine  the  value  of  the  constant  P,  which 
represents  the  pressure  exerted  upon  the  unit  of  surface,  and 
capable  of  producing  a  density  of  air  represented  by  unity, 
we  remark,  that  the  density  D'  at  the  lower  station  corres- 
ponds to  the  pressure  exerted  by  the  atmosphere  at  that 
point  :  this  pressure  is  measured  by  the  weight  of  a  column 
of  air  whose  base  is  the  superficial  unit,  and  whose  altitude 
is  equal  to  that  of  the  atmosphere  :  but  this  column  of  air  is 
equal  in  weight  to  the  mercurial  column  whose  height  is  h!  ; 
if  therefore  D"  denote  the  density  of  mercury,  the  mass  of 
the  column  will  be  expressed  by  1 X  A'D",  or  /i'D"  :  and  by 
multiplying  this  product  by  g^  we  shall  obtain  the  expression 
h'D"g,  for  the  weight  of  the  column  supported  at  the  lower 
station.  Such  will  be  the  pressure  capable  of  producing  the 
density  D'.  To  obtain  the  pressure  P  corresponding  to  the 
unit  of  density,  we  make  the  proportion 

D'  :  1  :  :  h'jy'g  :  P  ; 
whence, 


402  HYDROSTATICS. 

substituting  this  value  in  the  formula  (414),  there  results 
^=-ËF-Log- (415). 

719.  The  intensity  of  the  force  of  gravity  being  different 
at  di^rent  places  on  the  surface  of  the  earth,  the  weight  of 
the  same  column  of  mercury  will  likewise  vary  when  it  is 
transported  from  one  place  to  another  :  thus,  if  the  force  of 
gravity  be  denoted  by  g  at  on«  station,  and  by  (1  —^)g  at  a 
second,  the  mercurial  column  whose  height  is  h'  will  become 
heavier  or  lighter  at  the  second  station  than  it  was  at  the 
first,  according  as  ^  is  negative  or  positive. 

Let  the  quantity  ^  be  considered  positive  :  then  1  —  ^  will  be 
positive,  and  less  than  unity,  since  the  variations  of  gravity 
are  exceedingly  small.  But  a  column  of  mercury  whose 
height  is  h'  becoming  lighter  at  the  point  whose  gravity  is 
denoted  by  (1  — J)^,  it  will  correspond  to  a  less  pressure  of 
the  atmosphere,  and  hence,  the  density  of  the  air  correspond- 
ing to  this  pressure  will  be  less. 

The  densities  of  the  air  being  proportional  to  the  pressures 
exerted,  and  these  pressures  being  measured  by  the  weights 
of  the  column  of  mercury  whose  height  is  h'.  it  follows  that 
the  intensities  of  gravity,  which  are  represented  respectively 
by  g- and  (1 — <J)^  at  the  two  places,  will  be  proportional  to 
the  densities  corresponding  to  the  same  height  h'  of  the  mer- 
curial column  :  thus,  if  we  denote  by  d  the  density  of  the 
air  at  the  place  where  the  intensity  of  gravity  is  represented 
by  {\—^)g,  we  shall  have 

^:^(l-^)::D':c^; 
whence, 

This  value  of  the  density  must  be  substituted  in  the  for- 
mula (415),  in  order  that  it  may  become  applicable  to  the 
place  at  which  the  gravity  is  represented  by  {\—S)g:  the 
formula  will  thus  become 

-=DXW)Log^ (416). 

From  a  comparison  of  the  results  obtained  by  causing  pen- 
dulums to  oscillate  in  different  latitudes,  it  has  been  ascer- 


BAROMETER.  403 

tained  that  if  the  intensity  of  gravity  be  denoted  by  g  at  the 
latitude  of  45°,  the  quantity  ^  will  be  expressed  by  0.002837  X 
cos  2^,  when  the  latitude  is  supposed  to  become  equal  to  ■^. 
Hence,  by  substitution  in  the  preceding  formula,  we  obtain 

D"M/i'Log|-' 


D'(  1-0.002837  cos  2%/^) 
The  quantity  «5"  being  always  extremely  small,  we  may 

replace  t— r  in  equation  (416)  by  its  development  l+^S'+.J'^ 

i  — 0 

4-<5''+(fcc.  and  neglect  the  terms  ^^^  J 3,  dec.  as  extremely 
minute  with  reference  to  ^  :  we  thus  obtain- — -  =  1  +  «J'  ;  hence, 

1  — 0 

the  value  of  z  will  become 

;s= ^5^(1 +0.002837  cos  2^)  Log  ^ (417). 

720.  This  formula  has  been  obtained  upon  the  supposition 
that  the  temperature  remains  constant  in  passing  from  the 
lower  to  the  higher  station.  To  adapt  the  formula  to  the 
case  in  which  the  temperature  is  variable,  it  will  be  necessary  to 
know  the  law  according  to  which  air  expands  when  subjected 
to  a  change  of  temperature.  The  experiments  of  Gay  Lussac 
and  other  philosophers  demonstrate  conclusively  that  atmo- 
spheric air  when  perfectly  dry,  and  when  subjected  to  a  con- 
stant pressure,  expands  for  each  degree  of  Fahrenheit's 
thermometer,  between  the  temperatures  of  32°  and  212°,  ^^-^ 
of  its  volume  at  the  temperature  of  32°.  Thus  a  volume  of 
air  represented  by  unity  at  the  temperature  of  32°,  will  be- 

come  l  +  jôn  ^^''^^n   its    temperature    has    been    raised   to 

32°+/i°  :  and  since  the  densities  are  in  the  inverse  ratio  of 
the  spaces  occupied  by  the  same  mass,  it  follows  that  the 
density  d!  of  the  air,  at  the  temperature  32°  -h  n°,  will  be  ex- 
pressed by 


^480 
D'  being  the  density  at  the  temperature  32°. 

Cc2 


404 


HYDROSTATICS. 


The  coefficient  of  the  number  n  being-  very  small,  the  error 
which  will  be  introduced  by  assigning-  to  n  a  value  which 
shall  not  differ  greatly  from  its  real  value  will  always  be  ex- 
tremely small  ;  and  since  the  variations  in  temperature  which 
occur  in  passing  from  a  lower  to  a  higher  station  are  nearly 
uniform,  we  may,  without  sensible  error,  regard  the  temper- 
ature as  constant,  provided  we  assign  to  it  a  value  equal  to 
the  arithmetical  mean  between  the  temperatures  t  and  t'  at 
the  higher  and  lower  stations  ;  we  shall  thus  have 

and  the  density  d'  of  the  air,  which  was  previously  represented 
by  D',  will  become 

d'^ --J^ = ^ .    (418) 

^480V  2  '^^)  ^+  960 
But  the  density  of  mercury  bein^  increased  by  a  diminution 
in  the  temperature,  the  height  of  the  mercurial  column  at  the 
colder  station  will  be  less,  for  the  same  pressure,  than  it  would 
have  been  if  the  temperature  had  remained  constant,  and 
equal  to  that  at  the  warmer  station  ;  and  since  mercury  is 
found  to  expand  about  jy j,  P^^^  ^^  ^^^  h\i\k  for  every  degree 
of  Fahrenheit's  thermometer,  it  will  be  necessary  to  increase 
the  height  h,  which  is  supposed  to  correspond  to  the  colder 

station,  by  the  quantity  — —  taken  as  many  times  as  there 

are  degrees  of  difference  between  the  temperatures  of  the 
mercury  at  th«  two  stations,  in  order  to  reduce  this  height  h 
to  what  it  would  have  been,  if  the  temperature  had  remained 
constant,  and  equal  to  that  at  the  warmer  station.  Let  T 
and  T'  represent  the  temperatures  of  the  mercury  at  the  two 
stations  as  indicated  by  thermometers  in  contact  with  the 
barometers  ;  then  the  quantity  h  in  equation  (417)  should  be 
replaced  by 

^    ^(T--T). 
9742     ' 
hence,  by  substituting  this  value  for  h,  and  that  of  d'  (418) 
for  D',  in  formula  (417),  we  find 


BAROMETER.  466 


_D"M/i'  /    ,  i  +  i'-6'^\ 
-—fy-y-^     960     / 


h' 


X  (1+0.002837  cos  2^)  Log  — ^ 


K^+W") 


721.  Let  it  be  supposed  that  the  observations  which  deter- 
mine the  height  h'  are  made  in  the  latitude  of  45°,  and  at  the 
level  of  the  ocean  ;  we  shall  have 

cos  2\^=0  ; 

and  the  preceding  formula  will  give 

D'Mh'  z 


D'         /,  ,  ^-f  r-64\-  h' 


/,  .  <+r-64\- 


(419). 


a(i+ 


9742/ 

If  the  height  z  be  measured  trigonometrically,  and  the  quan- 
tities h,  h',  t,  i',  T,  T'  be  determined  by  taking  a  mean  result 
of  a  great  number  of  observations,  the  second  member  of  this 
equation  will  become  entirely  known,  and  therefore  the  con- 

M/i'D" 
stant  will  likewise  be  known.     This  constant  has  been 

thus  found  to  be  equal  to  60345  feet  :  if  its  value  be  substi- 
tuted in  that  ol  z,  we  shall  obtain  the  following  formula  : 

z=60345  ft.  (l+ttll^\  (1+0.002837  cos  2^) 

XLog  —, ^,_        (419  a). 

h(l+i — I) 
\        9742  / 

722.  The  second  member  of  this  equation  may  be  put 
under  a  more  convenient  form  ;  for,  we  have 

^±^=.001042(^+^-64)  ; 

and  if  we  denote  by  6  the  difference  between  the  temperatures 
T  and  T',  and  change  the  form  of  the  last  factor  in  equation 
(419  a),  there  will  result 


Log 


/       T--T\=^"g^'-^"g^-^"gC  +  9^)  ' 
A        9742/ 


406 


HYDROSTATICS. 


or,  by  developing  the  last  term  of  the  second  member,  retain- 
ing only  the  first  term  of  the  development,  we  shall  have 

in  which  M'  represents  the  modulus  of  the  system.     The 

numerical  value  of  the  coefficient  oîê  is  .000044.    Hence,  the 

equation  (419  a)  may  be  reduced  to 

5;=60345  ft.  [1  +  .001042(^+r-64)](l +.002837  cos  2^) 

X (Log  A' -Log  /i— .0000440). 

723.  To  apply  this  formula  to  the  determination  of  the  dif- 
ference of  level  of  two  stations,  it  will  be  necessary  to  observe 
carefully  the  altitude  of  the  mercurial  columns  at  each  station, 
and  the  temperature  of  the  atmosphere  as  indicated  by  a 
thermometer  placed  in  the  shade,  and  at  some  distance  from 
the  barometer.  The  temperature  of  the  mercury  as  shown 
by  a  thermometer  in  contact  with  the  tube  of  the  barometer 
should  likewise  be  noted.  These  observations  should  be 
made  at  the  same  instant,  by  different  observers,  at  the  two 
stations,  in  order  to  avoid  the  errors  which  might  arise  from 
a  change  of  pressure  or  temperature  during  the  interval  be- 
tween the  observations.  When  the  condition  of  simultaneous 
observations  becomes  impracticable,  it  will  be  advisable  to 
make  observations  at  one  of  the  stations,  the  lower  for  ex- 
ample, at  equal  intervals  before  and  after  the  time  of  observa- 
tion at  the  other  station.  Then,  an  arithmetical  mean 
between  the  first  and  last  results  may  be  considered  as  nearly 
equivalent  to  an  observation  made  at  the  instant  corresponding 
to  the  mean  between  the  two  times,  provided  the  interval  be 
but  short,  and  the  difference  between  the  results  of  the  two 
observations  inconsiderable. 

724.  The  general  formula  for  the  difference  of  level  of 
two  stations  having  been  obtained  upon  the  supposition  that 
the  atmosphere  is  in  equilibrio,  the  results  given  by  it  are  to 
be  relied  on  most  confidently  when  the  observations  have 
been  made  in  calm  weather. 


PART     FOURTH, 


HYDRODYNAMICS. 

OF  THE  DISCHARGE  OF  FLUIDS  THROUGH  HORIZONTAL  ORIFICES, 

725.  Experience  has  shown,  that  when  a  fluid  issues 
from  a  small  orifice  in  the  bottom  of  a  vessel,  the  superior 
surface  of  the  fluid  maintains  itself  in  a  position  sensibly  hori- 
zontal, during  the  discharge  of  the  fluid.  Hence,  if  we  con- 
ceive the  fluid  divided  into  horizontal  strata,  these  strata  may- 
be regarded  as  preserving  their  parallelism  during  their 
descent,  and  the  particles  may  be  considered  as  descending  in 
vertical  lines.  This  hypothesis,  however,  can  only  be  re- 
garded as  approximating  to  the  truth  ;  for,  if  the  form  of  the 
vessel  be  not  prismatic,  it  will  be  impossible  for  any  one 
stratum  to  occupy  the  place  of  that  immediately  beneath  it 
without  undergoing  some  change  in  its  dimensions  ;  its  par- 
ticles will  therefore  be  subjected  to  horizontal  motions.  More- 
over, the  particles  situated  in  the  immediate  vicinity  of  the 
orifice,  being  without  support,  yield  to  the  pressure  exerted 
against  them  by  the  adjacent  particles,  and  thereby  tend  to 
deflect  the  latter  from  their  vertical  directions  :  but,  in  what 
follows,  we  shall  omit  the  consideration  of  these  circum- 
stances, which  would  greatly  complicate  the  question,  and 
which  are  found  to  produce  but  a  slight  effect  when  the  form 
of  the  vessel  is  nearly  prismatic.  The  accuracy  of  the 
hypothesis  may  be  rendered  evident  by  mixing  with  the  fluid 
an  insoluble  powder  of  nearly  the  same  density  :  the  panicles 
of  this  powder  will  be  carried  along  with  the  fluid,  and  the 
paths  which  they  describe  may  be  readily  observed.  In  this 
manner  it  will  be  found  that  the  particles  descend  nearly  in 
vertical  lines  until  they  approach  very  near  to  the  orifice. 


408  HYDRODYNAMICS. 

726.  Let  it  be  supposed  that  the  ordinate  z  measures  the 
distance  mo  [Fig.  248)  of  one  of  the  fluid  strata  from  a  hori- 
zontal plane  AB,  which  will  be  assumed  as  coinciding  with 
the  surface  of  the  fluid.  The  form  of  the  vessel  being  deter- 
mined by  the  equation  of  its  interior  surface, /(^z-,  y,  2;,)=0, 
"we  can  deduce  from  this  equation  the  area  5  of  the  section 
which  corresponds  to  the  ordinate  z^  and  by  multiplying  this 
section  by  dz^  the  thickness  of  a  stratum,  we  shall  obtain  sdz 
for  the  volume  of  the  stratum.  This  being  premised,  it  is 
obvious  that  all  the  particles  composing  a  single  stratum  will 
have  a  common  velocity  ;  but  the  particles  of  different  strata 
will  have  different  velocities  ;  for,  the  fluid  being  supposed 
incompressible,  any  one  stratum  in  descending  through  the 
height  dz^  in  the  time  dt^  will  cause  a  volume  of  fluid  equal 
in  volume  to  the  stratum  to  issue  through  the  orifice.  But 
if  we  denote  by  u  the  velocity  of  the  fluid  at  the  orifice  EF, 
and  by  k  the  area  of  the  orifice,  the  space  described  by  a  par- 
ticle issuing  from  the  orifice,  in  the  time  dt^  will  be  expressed 
by  udt,  and  the  quantity  of  fluid  discharged  in  the  same  time 
will  be  represented  by  kiidt.  Equating  this  value  with  that 
of  the  stratum,  we  shall  obtain 

sdz^kudt (420)  ; 

whence, 

ku=s^ (421). 

dt  ^ 

727.  At  the  expiration  of  the  time  t,  the  velocity  of  the 

dz 
stratum  whose  section  is  s  will  be  equal  to  —,  and  if  this 

CtL 

velocity  be  represented  by  v,  the  equation  (421)  will  reduce  to 

ku=sv (422)  ; 

whence  we  conclude,  that  the  velocities  v  and  u  are  in  the 
inverse  ratio  of  the  sections  5  and  k.  This  result  might 
have  been  anticipated  ;  for,  the  velocity  must  evidently  in- 
crease in  the  same  ratio  that  the  area  of  the  section  is  dimin- 
ished, in  order  that  the  quantity  of  fluid  passed  through  the 
section  may  remain  constant. 

728.  At  the  expiration  of  the  time  t-\-dt,  the  velocity  v  will 

become  v-\--j-dt:  but  if  the  motions  of  the  particles  were 
dt 


DISCHARGE    OF    FLUIDS. 


409 


independent  of  their  action  upon  each  other,  the  incessant 
force  g^  which  sohcits  them,  would  communicate,  in  the 
instant  dt^  the  velocity  gdt  ;  hence,  the  velocity  lost  by  the 
stratum  whose  section  is  s,  and  velocity  o,  in  the  time  dl, 

will  be  expressed  by  gdt——dt,  and  consequently,  the  inces- 

dt 

sant  force  due  to  this  velocity  will  be  represented  by 

do 
^~dt' 

But  by  the  principle  of  D'Alembert,  an  equilibrium  would  sub- 
sist in  the  system  if  each  fluid  stratum  were  acted  upon  by 

the  force  lost  (g j  which  corresponds  to  it.     This  sup- 
position will  convert  the  equation  dp=J)gdz  (Art.  655)  into 
dp=B(^g-^j)dz (423). 

The  quantity  dp  represents  the  differential  of  the  pressure 
at  that  stratum  of  the  fluid  which  corresponds  to  the  ordi- 
nate z,  whilst  the  fluid  is  in  motion.  For,  the  force  g,  which 
acts  on  each  stratum,  being  resolved  into  two  components, 

one  of  which  j-  is  just  capable  of  producing  the  motion 

assumed  by  the  stratum,  the  other   component^— -5- will 

obviously  be  alone  eflective  in  producing  a  pressure  on  the 
other  strata  ;  and  the  expression  for  dp  has  been  obtained 
upon  the  supposition  that  these  second  components  v/ere 
alone  applied  to  the  fluid  particles. 

The  diflerential  dv  which  enters  into  the  preceding  equa- 
tion must  be  replaced  by  its  value  deduced  from  formula 
(422)  ;  from  that  formula  we  obtain 

v=— (424). 

s 

729.  The  second  member  of  this  equation  contains  the 
two  variables  ti  and  s  :  the  quantity  u,  which  expresses  the 
velocity  at  the  orifice,  is  a  function  of  the  time,  and  the  sec- 
tion 5  is  a  function  of  the  ordinate  z.  .~,^ 


410 


HYDRODYNAMICS. 


The  differential  of  v  regarded  as  a  function  of  z  expresses 
the  difference  between  the  velocities  of  two  consecutive  sec- 
tions, these  velocities  being  considered  at  the  same  instant. 
But  if  the  differential  be  taken  with  reference  to  ^  as  a  varia- 
ble, we  shall  obtain  the  difference  between  the  velocities  of 
two  consecutive  strata  which  pass  in  succession  through  the 
same  section  of  the  vessel.  And,  lastly,  if  we  wish  to  obtain 
the  difference  between  the  consecutive  velocities  of  the  same 
stratum,  we  must  differentiate  v  with  reference  to  the  two 
variables  t  and  z,  regarding  the  latter  as  a  function  of  the 
former. 

This  last  supposition  should  be  adopted  in  finding  the  value 
of  dv  as  employed  in  equation  (423).  We  shall  therefore 
differentiate  the  second  member  of  (424),  regarding  w  as  a 
function  of  t,  and  5  as  a  function  of  z,  which  is  itself  a  func- 
tion of  t.     But  the  differential  of  (424)  being  in  general 

dv—-du-\-kud-,, 
s  s 

or, 

,      kdu     ,       ds 
dv— ku  .  — , 

it  will  become,  when  modified  according  to  the  hypothesis 
assumed, 

,       k  du  ,,     ku  ^ds  ^dz 

s  dt         s^      dz     dt 

If  we  deduce  the  value  of  -7-  from  this  expression,  and  sub- 

dt 

stitute  it  in  equation  (423),  we  shall  obtain 

T-v  /     '       k  du  ,    ,  ku  ds   dz  J  \ 

dp  =  \J\gaz —  .  -^dz-\ .-—.-r-dz  I  : 

^         V®  s   dt  s^    dz  dt      / 

dt 

.,=D(,.._.f4>.-,4:). 

730.  This  equation  must  be  integrated  with  reference  to  z. 
We  remark,  however,  that  s  will  necessarily  vary  with  z,  but 

that  the  quantities  w  and  -^,  which  represent  the  particular 

dt 


eliminating  -^  by  means  of  equation  (420),  there  results 


DISCHARGE    OF    FLUIDS.  411 

values  of  v  and  -—  corresponding  to  the  orifice,  not  being 

functions  of  the  quantity  xr,  they  may  be  regarded  as  con- 
stant in  effecting  this  integration. 

731.  If  we  regard  u  and  —  as  constant,  it  is  obvious  that 

at 

all  the  integrals  will  be  taken  with  reference  to  z,  and  there- 
fore apply  merely  to  the  dimensions  of  the  vessel.     But,  when 

these  integrals  have  been  obtained,  we  may  regard  ii  and  — 

(JLt 

as  variables,  and  functions  of  t. 

732.  By  effecting  the  integration,  we  obtain 

^=K-^-4'/t-S^)+« (^^«)- 

The  velocity  u  which  enters  into  this  equation  is  equal  to 

dz 
the  value  —  corresponding  to  the  orifice,  and  will  obviously 

be  a  function  of  the  time.  Consequently,  as  the  quantity  u 
has  been  supposed  constant  in  the  preceding  integration,  the 
time  t  must  be  constant  likewise.  Hence,  the  constant  C 
will  in  general  be  a  function  of  the  time. 

733.  To  determine  this  constant,  let  P  represent  the 
pressure  sustained  by  the  superior  surface  CD  of  the  fluid 
{Fig.  249),  the  area  of  this  surface  being  denoted  by  s'.     If 

/dz 
—  be  taken  in  such  manner  that  it  shall  be 

equal  to  zero  when  s  becomes  equal  to  s\  this  section  s'  will 
correspond  to  an  ordinate  2;'=0L,  and  the  equation  (425) 
will  give,  upon  this  hypothesis, 

C=V-Ti(gz'-^^\. 

This  value  being  substituted  in  (425),  we  obtain 

,=P  +  d[,(._V)-..^/^^+,«.C^-|;)J....(426). 

734.  This  pressure  is  exerted  at  every  point  of  the  stratum 
whose  distance  from  the  plane  AB  is  equal  to  z.  If  we  wish 
to  obtain  the  pressure  Q,  at  the  orifice,  we  denote  by  z"  the 


412 


HYDRODYNAMICS. 


corresponding  value  of  the  ordinate  z  which  will  be  equal  to 
Ow,  and  observe  that  the  section  6-  will,  at  that  point,  be  equal 

/dz 
—  being  then  taken  between  the  hmits 

z=z'  and  z=z'\  we  shall  obtain,  by  representing  this  inte- 
gral by  N,  and  substituting  these  values  in  equation  (426), 

735.  This  equation  makes  known  the  pressure  at  the  ori- 
fice :  the  first  member  expresses  the  difference  between  the 
pressures  at  the  orifice  and  at  the  surface.  Let  these  pressures 
be  supposed  equal,  as  is  the  case  when  they  arise  from  the 
weight  of  the  atmosphere:  then,  d  — P  will  reduce  to  zero, 
the  common  factor  D  will  disappear,  and  there  Avill  remain 

gi,z"-z')-m^^^-\-\^i^  (^-l)  =0; 

but  the  area  k  of  the  orifice  being  always  supposed  less  than 

the  area  s'  of  the  superior  surface,  the  fraction  -^  will  be  less 

than  unity  ;  if  therefore,  we  wish  to  render  the  coefficient  of 
u"^  positive,  we  may  write  this  equation  under  the  form 

gi^^"^^')-m^^-\u^  {}-Ç)  =0 (427). 

736.  If  in  this  eqiiation  we  introduce  the  vertical  distance 
of  the  orifice  below  the  surface  of  the  fluid,  making 

z"-z'=h (428), 

we  shall  have 

The  quantity  h,  which  represents  the  distance  EP  (Pig.  250), 
will  be  constant  if  the  surface  of  the  fluid  be  supposed  to  be 
maintained  at  the  same  height  ;  but  it  will  be  variable  if  the 
vessel  be  supposed  to  discharge  its  contents  without  being 
replenished. 

737.  In  the  latter  case,  if  we  make  EO=a,  P0=5r,  and 
EP=r/j,  we  shall  have  the  relation 

h  =  a-z (430)  ; 


DISCHARGE   OF   FLUIDS.  413 

and  the  equation  (429)  will  become 

g{a-z)-m'^-^-iu^  (l-^)  =0 (431). 

738.  If  the  surface  of  the  fluid  be  constantly  maintained 
at  the  same  height,  the  quantity  h  will  have  a  constant  value, 
and  the  integral  N,  which  will  then  be  a  function  of  constant 
quantities,  will  likewise  be  invariable.  Thus,  equation  (429), 
containing  no  other  variables  than  t  and  m,  may  be  put  under 


the  form 


a — b~ cu^  =0 

dt  ' 


from  which  we  deduce 

hdu 


dt—- 


a — ca^ 

This  equation  can  be  readily  integrated  by  the  method  of 
rational  fractions;  for,  if  we  make  h=h'c^  and  a=a'^c,  the 
quantity  c  will  become  a  factor  of  the  numerator  and  de- 
nominator, and  may  be  stricken  out  ;  whence  we  obtain 

,^        h'du 

dt= . 

The  second  member  of  this  equation  being  resolved  into 
factors,  we  shall  have 

,^  2-^^r2^'^" 


a'-\-u     a' — u 
which,  being  integrated,  gives 


or, 


^=^  log  {a'+u)-^,  log  {a'-u)-hC', 


.      b'  ,      a'+u  ,  ^ 

t=—-  log — ■ hC. 

2a'     ^a'—u 


The  constant  C  is  determined  by  the  condition  that  the  ve- 
locity u  is  equal  to  zero  at  the  same  instant  as  the  time  t  ; 
thus,  the  supposition  of  w=0,  and  i=0,  reduces  the  preceding 
equation  to 

^'og  1+0=0; 


414  HYDRODYNAMICS. 

or, 

C=0: 
whence, 

6'  ,      a'-\-u 

this  equation  will  determine  u,  if  we  suppose  the  time  Mo  be 
given. 

739.  If  we  denote  by  e  the  base  of  the  Naperian  system, 
and  pass  from  logarithms  to  numbers,  we  shall  obtain 

2a't 

a'-\-u       -^r. 

-, =e 

a — u 

and  by  resolving  the  equation  with  reference  to  w,  there 
results 

2a't 

_  «te '''-!). 

or  replacing  a'  and  h'  by  their  values  (Arts.  737  and  738),  the 
expression  for  the  velocity  will  become 


But,  if  the  area  of  the  orifice,  which  is  denoted  by  A-,  be  sup- 
posed extremely  small,  the  exponent  of  e,  increasing  with  the 
time  ^,  will  become  exceedingly  great  after  the  expiration  of  a 
very  short  time.  Hence,  we  may  neglect  unity  in  the  nume- 
rator and  denominator  of  the  last  factor,  as  very  small  with 
reference  to  the  term  which  precedes  it,  and  the  value  of  u 
will  then  be  reduced  to 


vte) 


by  neglecting  k^  with  reference  to  s'^. 

Thus,  it  appears  that  the  expression  */(^gh)  is  a  limit 
which  the  velocity  of  the  fluid  at  the  orifice  never  attains,  but 
to  which  this  velocity  becomes  very  nearly  equal  after  the 
expiration  of  an  exceedingly  short  time. 


DISCHARGE    OF   FLUIDS.  415 

The  value  of  the  velocity  being  thus  determined,  we  sub- 
stitute it  in  equation  (425),  and  thence  deduce  the  pressure  on 
the  unit  of  surface. 

740.  If  the  vessel  be  supposed  to  empty  itself,  the  upper 
surface  will  be  depressed  as  the  fluid  is  discharged,  and  the 
quantity  A,  or  (a  —  z)  must  therefore  be  regarded  as  variable 
in  equations  (429)  and  (431). 

The  equation  (431)  will  thus  contain  the  three  variables 
t,  u,  and  z,  and  will  consequently  be  insufficient  for  the  solu- 
tion of  the  problem  :  but  a  second  relation  may  be  obtained 
by  means  of  equation  (420)  in  which  we  replace  *  by  s',  and 
thus  obtain 

ku=s'^ (432). 

741.  This  equation  likewise  contains  three  variables,  and 
we  are  therefore  unable  to  integrate  it  ;  but  it  will  serve  to 
eliminate  z.  For  this  purpose,  we  differentiate  equation 
(431),  which  gives 

—g- A;N- u-r\  1 r-  I  =0> 

^dt  dV-         dt\       5'V        ' 

dz 
and  by  eliminating  — ^,  we  obtain 

(jLZ 

~g—, — ^^N-- — ?«-—(  1——  )  =0. 
s'  di''         dt  \       s'V 

This  equation,  which  can  only  be  integrated  by  approxima- 
tion, makes  known  the  relation  between  the  time  and  the 
velocity. 

742.  When  the  orifice  is  supposed  extremely  small,  the 
terms  containing  k  may  be  neglected,  and  the  equation  (426) 
will  be  reduced  to 

p=V-\.-Dg{z-z'); 
but  z~z'  is  represented  by  On — OL  {Fig.  249)  ;  and  it  will 
therefore  express  the  distance  of  the  point  w,  whose  ordinate 
is  equal  to  z,  beneath  the  surface  of  the  fluid.  Hence,  the 
pressure  p  exerted  upon  the  unit  of  surface  at  the  point  n  is 
equal  to  the  pressure  P  at  the  surface  of  the  fluid,  plus  the 
pressure  arising  from  a  column  of  a  fluid  whose  height  is 
equal  to  the  distance  of  this  point  below  the  surface. 


416  HYDRODYNAMICS. 

It  should  be  remarked,  that  this  pressure  is  precisely  that 
which  would  be  exerted  at  the  point  n  if  the  fluid  were  sup- 
posed at  rest. 

743.  If  the  terms  containing  k  in  equation  (429)  be  neg- 
lected as  infinitely  small,  it  will  reduce  to 

whence, 

u=^{2gh) (433): 

and  we  therefore  conclude,  that  when  a  fluid  escapes  from  an 
infinitely  small  orifice  in  the  bottom  of  a  vessel,  the  velocity 
will  be  the  same  as  that  acquired  by  a  heavy  body  in  falling 
through  a  distance  equal  to  the  height  of  the  surface  of  the 
fluid  in  the  reservoir  above  the  orifice  ;  and  since  it  has  been 
shown  (Art.  405)  that  a  body  projected  vertically  upwards 
will  rise  to  a  height  equal  to  that  through  which  it  must  fall 
to  acquire  the  velocity  of  projection,  it  follows,  that  if  by 
means  of  a  curved  tube,  the  jet  of  fluid  be  directed  upwards, 
it  will  rise  to  the  level  of  the  surface  of  the  fluid  in  the 
reservoir. 

744.  The  expression  for  the  velocity  with  which  a  fluid 
will  issue  from  an  extremely  small  orifice  in  the  bottom  of  a 
vessel  may  be  investigated  in  a  more  elementary  manner,  as 
follows.  Let  EF  {Fig.  251)  represent  a  very  small  orifice 
in  the  bottom  of  a  vessel  ABCD,  which  is  filled  with  a  fluid 
to  the  level  AB,  and  let  GF  represent  an  infinitely  thin  stra- 
tum of  the  fluid  directly  above  the  orifice  EF.  Denote  the 
height  of  this  stratum  by  dh,  the  entire  height  of  the  fluid 
FI  being  represented  by  h.  Then  if  the  stratum  of  fluid  GF 
be  supposed  to  fall  through  the  height  HF  under  the  influ- 
ence of  the  force  of  gravity,  it  will  acquire  a  velocity  v,  ex- 
pressed by 

v  =  ^{2gy.YYL)^^{2gXdh). 

But  if  the  stratum  be  supposed  to  descend  through  the  same 
height,  being  urged  by  its  weight  and  the  pressure  arising 
from  the  column  of  fluid  GI,  which  is  directly  over  it,  the 
incessant  force  g',  which  is  then  exerted  upon  it,  will  be  to 
the  force  of  gravity,  as  FI  to  FH.    Hence,  we  shall  have 

^~FH    dfi 


DISCHAKGK    OF    FLUIDS.  417 

Again,  if  u' denote  the  velocity  acquired  by  tne  stratum  in 
descending  through  the  space  FH,  when  urged  by  the  force 
g\  we  shall  have 

and  by  comparing  this  value  with  that  of  v^  we  find 

V      ^(2gxdh)' 
or,  by  substituting  the  value  of  — ,  and  reducing,  there  results 

This  expression  is  precisely  the  same  as  that  which  would 
be  obtained  for  the  velocity  of  a  body  falling  freely  through 
the  height  FI. 

745.  When  the  orifice,  which  is  still  supposed  exceed- 
ingly small,  is  pierced  in  the  vertical  face  of  a  vessel,  the 
fluid  will  issue  in  a  horizontal  direction,  and  will  describe  the 
arc  of  a  parabola,  if  the  resistance  of  the  air  be  neglected.  The 
angle  of  projection  denoted  by  a  in  equation  (289),  being  in 
the  present  case  equal  to  zero,  we  shall  have  tang  «=0, 
cos  ct=l:  these  suppositions  reduce  the  formula  (289)  to 

an  equation  of  a  parabola  whose  axis  is  vertical,  and  whose 
vertex  coincides  with  the  origin  of  co-ordinates. 

746.  The  distance  to  which  the  fluid  will  spout  upon  a 
horizontal  plane  situated  at  any  distance  below  the  orifice 
may  be  readily  determined.  For,  let  O  {Fig.  252)  represent 
an  orifice  in  the  vertical  side  of  a  vessel  which  is  filled  with  a 
fluid  to  the  level  EF  ;  and  let  AB  represent  the  horizontal 
plane  upon  which  the  jet  is  allowed  to  fall.  Then,  the  quan- 
tity h  will  represent  the  distance  OF,  and  the  ordinate  CD  of 
the  parabola  OD  will  be  determined  by  making  y=OC  :  we 
thus  obtain 

CD=:r  =  v'(4%)=V(OFxOC). 

But  the  expression  y'(OFxOC)  is  equal  to  the  ordinate  OG 
of  a  semicircle  described  upon  CF  as  a  diameter.  Hence, 
we  derive  the  following  rule  :  The  horizontal  distance  to 
wliicli  ajluid  will  spout  from  an  orifice  in  the  vertical  side  of 

Dd 


418  HYDRODYNAMICS. 

a  vessel,  is  equal  to  double  the  ordinate  of  a  semicircle  de- 
scribed upon  the  distance  intercepted  between  the  upper  sur- 
face of  the  fluid  and  the  horizontal  plane  iipon  which  the 
fluid  falls  ;  this  ordinate  being  drawn  through  the  point 
lohich  corresponds  to  the  orifice. 

When  the  orifice  is  pierced  at  the  middle  of  the  hne  CF, 
the  ordinate  OG  will  be  a  maximum,  and  the  distance  to 
which  the  fluid  will  spout  will  therefore  be  the  greatest. 

747.  The  velocity  u  having  been  determined,  we  can 
readily  ascertain  the  quantity  of  fluid  discharged  in  the  time  t. 
For  this  purpose,  we  remark,  that  whilst  the  stratum  of  fluid 
CD  {Pig.  250)  sinks  to  the  level  MN,  a  volume  of  fluid 
equal  to  that  contained  between  the  planes  CD  and  MN  must 
pass  through  the  orifice.  But  if  we  represent  by  5  a  section 
of  the  vessel,  and  by  dz  the  thickness  of  an  elementary 
stratum,  the  integral  fsdz  taken  between  limits  CD  and  MN 
will  express  the  volume  of  fluid  discharged.  If  this  volume 
be  denoted  by  Q,  we  shall  have 

Q,=fsdz (434)  : 

but  the  equation  (420)  gives 

sdz=kudt  ; 
whence,  by  substitution,  we  obtain 

Q,  =fkudt. 
The  value  of  the  quantity  discharged  may  be  deduced  imme- 
diately from  that  of  the  velocity.     For,  if  de  represent  the 
space  passed  over  by  the  fluid  filament  in  the  time  dt,  upon 
leaving  the  orifice,  we  shall  have 

udt=de: 
and  if  this  expression  be  multiplied  by  k,  the  area  of  the  ori- 
fice, we  shall  obtain  kudt  for  the  volume  discharged  in  the 
time  dt.     Taking  the  integral  fkudt,  we  shall  find  the  quan- 
tity discharged  in  the  time  t. 

To  effect  the  integration,  we  replace  u  by  its  value  y/{2gh) 
given  in  equation  (433)  :  we  thus  find 

Qi=k^{2g)f^h.dt (435). 

748.  Two  distinct  cases  may  now  be  presented,  viz.  when 
h  is  constant,  and  when  h  is  variable.     The   first  occurs 


DISCHARGE   OP    FLUIDS.  419 

when  the  fluid  in  the  reservoir  is  constantly  maintained  at 
the  same  height,  and  the  preceding  equation  can  then  be 
integrated  without  difliculty,  since  the  quantity  h  may  be 
replaced  by  a  constant  a. 
Thus,  we  shall  have 

Q,=kt^{2ga)  +  C. 
The  constant  C  may  be  determined  by  the  condition  that  the 
quantity  Q,  is  equal  to  zero  at  the  commencement  of  the  time, 
or  Q,=Oj  and  ^=0  ;  hence, 

and  the  equation  therefore  reduces  to 

a=A-V(2^«) (436). 

749.  If  the  orifice  k  be  supposed  circular,  its  radius  being 
represented  by  r,  we  shall  have 

k—vr^\ 
and  the  formula  will  become 

a=^^(2^)^rV« (437). 

The  quantity  '^^/{2g)  will  be  the  same  for  all  problems  which 
may  be  proposed,  and  its  value  may  be  immediately  deduced, 
since  we  have 

^=3.14159,    ^=32.1.598. 
The  quantity  g  being  expressed  in  feet,  the  values  of  r  and  a 
must  be  expressed  in  units  of  the  same  kind,  and  the  quan- 
tity discharged  will  then  be  expressed  in  cubic  feet. 

750.  The  time  t  must  be  expressed  in  seconds,  since  the 
second  has  been  adopted  as  the  unit  of  time  in  determining 
the  value  of  g. 

7o\.  If  the  fluid  be  water,  the  weight  of  the  quantity  dis- 
charged may  be  determined  by  allowing  62ilbs  for  every 
cubic  foot. 

752.  The  formula  (437)  likewise  serves  to  determine  the 
time  necessary  for  a  given  quantity  of  fluid  to  be  discharged 
from  an  orifice  in  a  vessel,  when  the  fluid  is  maintained  at  a 
constant  height  ;  for  the  formula  gives 

t= ^_ (438). 


420 


HYDRODYNAMICS. 


753.  As  an  example,  let  the  vessel  be  supposed  cylindrical, 
the  radius  of  its  base  being  denoted  by  b  ;  and  let  it  be  re- 
quired to  determine  the  time  necessaiy  to  discharge  a  volume 
of  fluid  equal  to  that  of  the  cylinder. 

In  this  case,  the  horizontal  sections  being  all  equal  to  xb^, 
the  equation  (434)  will  give 

Çi^firb^  dz  ; 
and  consequently, 

a^Trb^z+C. 
Taking  the  integral  between  the  limits  «=0  and  Zz=a,  there 
results 

(X=vb^a. 
This  value  substituted  in  formula  (438)  gives 

Trab^ 


t= 


or,  by  reduction. 


b^^a 


t= 


754.  If  we  suppose  the  fluid  to  be  maintained  at  a  height 
a'  in  a  second  vessel,  and  denote  by  Q,'  the  quantity  dis- 
charged from  an  orifice  k'  in  the  time  t,  the  equation  (436), 
when  applied  to  the  present  case,  will  give 

Ci'=k'^{2g).  t^a'  ; 
and  by  comparing  this  equation  with  (436),  we  can  estabhsh 
the  proportion 

Gl:  a'  ::  k^{2g).t^a:  k'^{2g).t^a' ; 
or,  by  suppressing  the  common  factor  t-^{2g),  this  proportion 
becomes 

d  :  Q,'  ::  k^a:  k'^/a'. 
Hence  it  appears,  that  the  quantities  discharged  in  the  same 
time,  from  orifices  of  different  sizes,  and  situated  at  different 
depths,  are  directly  proportional  to  the  areas  of  those  orifices 
and  the  square  roots  of  their  deptJis. 

755.  From  the  formula  (436)  we  can  deduce  another  con- 
venient theorem  relative  to  the  quantity  of  fluid  discharged. 
For,  let  s  represent  the  space  through  which  a  body  would 
fall  in  the  time  t  ;  we  shall  have 


DISCHARGE    OF    FLUIDS.  421 

or, 

Substituting  this  value  for  t  in  equation  (436),  we  obtain 

and  since  ■^/{as)  is  equal  to  a  mean  proportional  between  tlie 
distances  a  and  s,  we  deduce  the  following  rule  :  The  volume 
of  fluid  discharged  from  an  oriflce  k,in  the  time  t,is  equal 
to  tioice  the  volume  of  a  cylinder  whose  base  is  the  area  of 
the  oriflce,  and  whose  height  is  a  mean  projjortional  between 
the  depth  of  the  oriflce  beloiv  the  surface  and  the  distance 
through  which  a  body  u'ouldfall  in  the  time  t. 

756.  Let  the  vessel  be  now  supposed  to  discharge  itself, 
without  receiving  an  additional  supply  of  fluid  :  the  quantity 
h  in  equation  (433)  must  then  be  regarded  as  variable,  and 
being  replaced  by  [a — s),  that  equation  will  become 

This  value  of  u  substituted  in  (432)  gives 

fif  —         *'^^ 

~k^[2^^^a-z)y 

or, 

dt^  ^'^^ (439). 

The  quantity  s'  represents  the  section  of  the  vessel  which 
corresponds  to  the  upper  surface  of  the  fluid.  This  section 
will  be  a  function  of  the  variable  z,  and  may  be  eliminated  by 
means  of  the  equation  of  the  interior  surface  of  the  vessel. 
Thus,  the  value  of  s'  in  terms  of  z  being  introduced  into 
equation  (439)  will  render  that  equation  susceptible  of  inte- 
gration, and  the  relation  between  z  and  t  will  therefore 
become  known.  If  we  subtract  the  value  of  z  thus  obtained 
from  the  constant  a,  we  shall  obtain  an  expression  for  h  in 
terms  of  t,  which  substituted  in  (435)  will  give,  after  integra- 
tion, a  relation  between  the  time  t  and  the  quantity  dis- 
charged Q,. 

757.  Let  us  take,  as  an  example,  a  vessel  whose  interior 
surface  has  the  form  of  a  paraboloid  of  revolution.     T!  is 

36 


422  HYDRODYNAMICS. 

surface  being  generated  by  the  revolution  of  the  parabohc 
arc  AD  {Mg.  253)  about  the  vertical  axis  AB  ;  if  we  denote 
by  a  the  distance  AB  between  the  orifice  and  the  surface  of 
the  fluid  in  its  primitive  position,  by  z  the  distance  PB, 
and  by  y  the  ordinate  PM,  we  shall  have  the  relation, 

y^=1){a—z) the  equation  of  a  parabola  referred  to  its 

vertex  A. 
Hence,  if  ?r  represent  the  ratio  of  the  circumference  to  the 
diameter,  the  area  of  the  circle  described  with  the  radius  PM 
will  be  expressed  by  jry^  =7rp(a — z)  ;  and  consequently, 

s'=^p{a-z) (440). 

Let  this  value  be  substituted  in  (439),  and  we  shall  obtain 

dt=-, — ^— —  X — -, ^dz  ; 

or,  by  reduction, 

dt  =  :r^^'^---(a-zydz. 

758.  For  the  purpose  of  integrating  this  equation,  we  make 
a—z=x]  whence, 

f{a—zYdz=—fx^dx=—%x^  +  C'. 
replacing  x  by  its  value,  we  have 

f{a-zYdz  =  -%{a-z)^  +  C', 
and  consequently. 

The  constant  C  is  determined  by  making  2;=0andi=0; 
this  supposition  gives 


and  the  equation  (441)  can  therefore  be  reduced  to 


vp 


■[a^—{a-zY]. 


'k^{2gy 

To  determine  the  quantity  discharged  in  a  given  time,  we 
find  in  this  equation  the  value  of 


■.-.^{J-^>) 


t=-^^y.^     ''^ 


DISCHARGE    OF    FLUIDS.  423 

and  substitute  it  for  h  in  formula  (435)  :  we  thus  obtain  the 
relation 

This  equation  may  be  integrated  by  a  process  entirely  similar 
to  that  adopted  in  finding  the  relation  between  z  and  t. 

759.  Let  it  be  required  to  determine  the  time  in  which  the 
water  contained  in  a  vessel  having  the  form  of  a  right  cyl- 
inder will  be  discharged  through  an  orifice  in  the  bottom  of 
the  vessel.  Let  h  represent  the  radius  of  a  section  of  the 
cylinder  by  a  plane  perpendicular  to  its  axis  :  then,  s'=?rb'', 
and  the  equation  (439),  when  applied  to  the  present  case,  wil. 
give 

Making  a—z=.T,  then  integrating  the  transformed  equation, 
and  replacing  a:  by  its  value,  we  find 

The  constant  is  determined  as  in  the  last  example,  by  making 
z=0  and  t=0  :  whence  we  deduce 

The  integral  being  taken  between  the  limits  z=0  and  z=a, 
we  find,  for  the  time  of  emptying  the  vessel, 

'-iBm^" <^*^'- 

If  we  suppose,  as  in  Art.  749,  that  the  orifice  is  a  circle  whose 
radius  is  equal  tor,  we  shall  have  k=7rr^  :  this  value  reduces 
(443)  to 

By  comparing  this  result  with  that  obtained  in  Art.  753,  it 
will  appear  that  the  time  necessary  for  the  entire  discharge 
of  the  fluid  when  the  vessel  empties  itself,  is  double  that  in 
which  an  equal  quantity  of  fluid  would  flow  through  the 
same  orifice  if  the  vessel  were  kept  constantly  full. 

760.  The  formulas  (442)  and  (443)  will  serve  as  a  guide  in 


424  HYDRODYNAMICS. 

the  construction  of  a  clepsydra,  or  water-clock.  This  instru- 
ment consists  merely  of  a  vessel  from  which  the  water  is 
allowed  to  escape  through  an  orifice  in  the  bottom,  and  the 
intervals  of  time  are  measured  by  the  depressions  of  the 
upper  surface.  Thus,  if  we  wish  the  clock  to  run  12  hours, 
we  reduce  the  time  o  seconds,  which  g-ives  12 x (60) 2,  or 
12  X  3600  ;  and  by  substituting  this  value  of  t  in  formula 
(443),  we  can  then  assume  arbitrarily  two  of  the  three  quan- 
tities A-,  6,  and  a.  Let  the  values  of  k  and  h  be  assumed  ; 
that  of  a,  the  height  of  the  clepsydra,  will  then  result  from 
formula  (443). 

To  discover  the  manner  in  which  this  height  should  be 
divided  in  order  that  the  superior  surface  of  the  fluid  may 
be  depressed  through  the  several  divisions  of  the  scale  in 
equal  intervals  of  time,  we  deduce  from  equation  (442)  the 
value  of  {a—z),  which  is 

.A•V(2^)^^ 


-H^'^-'W) 


and  by  making  t  successively  equal  to  1  hour,  2  hours,  3 
hours,  &c.,  we  can  determine  the  corresponding  values  of 
a—z,  which  should  be  laid  off  from  the  bottom  of  the  vessel. 
We  can,  however,  readily  discover  the  general  law  according 
to  which  the  scale  must  be  divided  :  for,  since  the  vessel  is 
supposed  to  discharge  itself  in  12  hours,if  we  make  ^=12  hrs., 
we  shall  have  a—z=0;  and  consequently, 
/•      Â:(12hrs.)v^(2,^)_^. 

or, 

(12hrs.)^^'(2o-) 

2^6=*  '  ^ 

When  ^=11  hrs,,  we  have 

t .  V(2^)     (llhrs.)V(2g-)_,, 

and  therefore, 

a-z  =  {^/a-\i^ar-  =^(j\y  Xa. 
In  like  manner,  when  ^=10  hrs,  we  shall  find 
a—z=:{j%y-  xa. 


DISCHARGE   OF   FLUIDS.  425 

Thus  the  successive  values  of  a—z^  which  correspond  to  the 
several  hours,  will  bear  to  each  other  the  same  relations  as  the 
terms  in  the  series 

(TV)^(T^3)^(^)^&c. 

These  terms  are  to  each  other  in  the  same  ratio  as  the 
squares  of  the  natural  numbers  1,  2,  3,  &.C.  Hence,  if  we 
divide  the  whole  height  a  into  144  equal  parts,  and  lay  off 
from  the  bottom  of  the  vessel  distances  which  shall  be  equal 
respectively  to  1,  4,  9,  &.c.  of  these  parts,  we  shall  obtain  the 
points  of  division  in  the  scale  which  will  correspond  to  the 
upper  surface  of  the  fluid  at  the  expiration  of  the  several 
hours.  The  form  of  the  vessel  being  prismatic,  the  figure 
of  its  base  may  be  assumed  arbitrarily. 

761.  When  the  surface  of  the  fluid  shall  have  arrived 
nearly  at  the  bottom  of  the  orifice,  the  quantity  discharged 
will  be  influenced  by  the  formation  of  a  hollow  tunnel,  which 
is  then  found  to  be  produced  directly  above  the  orifice  :  it  is 
therefore  advisable  to  employ  only  the  first  eleven  divisions 
of  the  scale. 

762.  It  usually  occurs  that  the  condition  of  the  particles 
descending  in  vertical  lines,  and  with  velocities  which  are 
equal  at  every  point  of  the  same  stratum,  ceases  to  be  ful- 
filled when  the  surface  of  the  fluid  has  arrived  within  4  or  5 
inches  of  a  horizontal  orifice.  The  fluid  particles  then 
assume  directions  which  are  more  or  less  inclined  to  the  hori- 
zon, and  the  tunnel  spoken  of  in  the  last  article  is  then 
formed.  When  the  orifice  is  found  at  a  considerable  depth, 
the  upper  surface  of  the  fluid  remains  sensibly  horizontal, 
and  the  tunnel  above  the  orifice  is  no  longer  formed,  in  con- 
sequence of  the  greater  velocity  with  which  the  fluid  par- 
ticles near  the  orifice  are  compelled  to  flow  into  the  vacancy 
which  has  been  left  by  those  immediately  preceding  them. 

763.  This  tunnel  becomes  much  less  perceptible  when  the 
orifice  is  formed  in  the  side  of  a  vessel.  But  when  the  upper 
surface  of  the  fluid  has  nearly  attained  the  level  of  the  orifice, 
a  slight  depression  on  the  side  of  the  orifice  begins  to  be 
observed. 

764.  This  tendency  of  the  fluid  particles  towards  the  ori- 
fice, occasioned  by  their  sustaining  less  pressure  in  that  direc- 


426  HYDRODYNAMICS. 

tion,  gives  rise  to  a  contraction  in  the  jet  of  fluid,  which,  in 
issuing  from  the  orifice,  assumes  the  form  of  a  truncated 
pyramid  or  cone,  whose  greater  base  corresponds  to  the  ori- 
fice. This  diminution  in  the  size  of  the  jet  is  called  the  con^ 
traction  of  tJie  vein. 

With  a  circular  orifice,  the  smallest  section  of  the  fluid  vein 
is  found  at  a  distance  from  the  orifice  equal  to  the  radius  of 
the  orifice.  Beyond  this  point  the  diameter  of  the  section 
again  increases,  so  that  the  entire  jet  has  the  form  of  two 
truncated  cones  which  are  united  by  their  smaller  bases. 

765.  The  contraction  of  the  vein  likewise  takes  place 
when  the  orifice  is  pierced  in  the  side  of  a  vessel  ;  but  if  the 
orifice  be  large,  and  be  placed  at  a  short  distance  below  the 
surface  of  the  fluid  in  the  reservoir,  the  jet  will  be  found  to 
be  more  contracted  in  the  vertical  than  in  the  horizontal 
direction. 

766.  When  a  conical  tube  whose  interior  surface  corres- 
ponds to  the  form  of  the  contracted  vein  is  adapted  to  an 
orifice  pierced  in  a  thin  plate,  the  quantity  discharged  is  found 
to  be  very  nearly  the  same  as  though  the  fluid  issued  directly 
through  the  orifice.  Hence,  we  may  regard  the  vessel  as 
coiitinued  to  the  point  at  which  the  greatest  contraction  of 
tiie  stream  takes  place,  and  consider  the  least  section  as 
forming  the  real  orifice. 

It  is  proved  by  experience,  that  the  quantity  actually  dis- 
charged may  be  deduced  from  that  calculated  according  to 
the  theory,  by  sinjply  changing  the  value  of  the  constant  k. 
Thus,  if  we  represent  by  MA:  the  area  of  the  orifice  which  has 
been  calculated  from  a  knowledge  of  the  quantity  actually 
discharged,  tne  theoretic  formula 

must  be  modified  by  substituting  Wt  for  k  :  we  shall  thus 
obtain,  for  the  actual  discharge, 

767.  When  the  orifices  are  pierced  in  thin  plates,  the  ratio 
M  is  found  to  be  independent  of  the  size  of  the  orifice,  and  of 
its  depth  below  the  surface,  provided  that  depth  be  not  very 
small.    Hence,  if  we  represent  by  Q,'  the  quantity  discharged 


DISCHARGE    OF   FLUIDS.  427 

from  an  orifice  k'  at  the  depth  a\  we  shall  have  the  pro- 
portion 

a  :  a'  :  :  Wc^{2g) .  V«  :  MAV(2^)  •  V«'  i 
and  we  therefore  conclude,  that  the  quantities  discharged  from 
two  such  orifices  are  to  each  other  as  the  products  of  the  areas 
of  those  orifices,  and  the  square  roots  of  their  depths. 

768.  The  number  M  has  been  found  by  Bossut  to  be  about 
0.62,  and  the  orifice  k  must  therefore  be  multiplied  by  this 
fraction,  in  order  that  the  quantity  given  by  the  formula  may 
correspond  with  the  results  of  experiment.  Thus,  the  cor- 
rected expression  for  the  quantity  discharged  will  be 

Q  =  (0.62)AV(2^).V«- 
This  formula  is   alike   applicable,  whether   the   orifice   be 
pierced  in  the  side  or  bottom  of  a  vessel. 

769.  When  the  vessel  is  allowed  to  empty  itself,  the  cir- 
cumstances of  the  discharge  become  very  complicated  after 
the  upper  surface  of  the  fluid  has  fallen  to  within  a  short  dis- 
tance of  the  orifice.  If,  however,  we  only  consider  the  ex- 
penditure previous  to  the  arrival  of  the  upper  surface  within 
a  few  inches  of  the  orifice,  the  same  correction  may  be 
applied  to  formula  (439),  which  will  thus  become 

-(0.62)Av(2^)v/(«-^)' 
and  will  serve  to  determine  the  time  necessary  for  a  given 
quantity  of  fluid  to  be  discharged. 

770.  In  applying  the  preceding  correction  to  the  theoreti- 
cal discharge,  it  has  been  supposed  that  the  orifice  was 
pierced  in  a  thin  plate  :  when  a  similar  orifice  is  pierced  in  a 
thick  plate,  the  quantity  discharged  is  found  to  be  consider- 
ably greater.  Hence  it  occurs,  that  when  the  fluid  is  dis- 
charged through  a  thick  plate,  or  through  a  cylindrical  tube 
applied  to  the  orifice,  the  coefiicient  0.62,  which  has  been  em- 
ployed in  calculating  the  discharge  through  a  thin  plate,  is 
no  longer  applicable.  In  this  case  the  fluid  adheres  to  the 
sides  of  the  tube,  and  the  contraction  of  the  stream  is  in  a 
great  measure  avoided.  The  lengths  of  such  tubes,  accord- 
ing to  Bossut,  should  be  at  least  twice  the  diameter  of  the 
orifice,  in  order  that  the  contraction  of  the  vein  may  be  pre- 


428  HYDRODYNAMICS. 

vented.  There  will  however  be  a  limit  to  the  length,  proper 
to  be  given  to  such  tubes,  since  the  friction  of  the  fluid 
against  the  sides  of  the  tube  will  necessarily  increase  with 
its  length. 

771.  The  quantities  discharged  by  cylindrical  tubes  are 
proportional  to  the  products  of  the  orifices  by  the  square  roots 
of  their  depths,  as  in  the  case  of  apertures  pierced  in  a  thin 
plate  ;  but  the  coefficient  M,  by  which  the  area  of  the  orifice 
must  be  multiplied  for  the  purpose  of  reducing  the  theoretical 
discharge  to  that  given  by  experiment,  has  been  found  by 
Bossut  to  be  about  ||,  or,  more  accurately,  0.81,  when  a  short 
cylindrical  tube  is  applied  to  the  orifice.  Thus,  the  formula 
(436),  which  serves  to  determine  the  quantity  discharged  from 
a  reservoir  in  which  the  fluid  is  maintained  at  a  constant 
height,  will  become,  when  corrected  for  the  case  of  a  cylin- 
drical tube, 

a=(0.Sl)^(2^).AV«; 
or,  if  we  replace  k  by  its  value  «-/-s,  r  denoting  the  radius  of 
its  section,  the  formula  may  be  written 

a=(0.81)^(2o-).^r=V«- 

772.  When  the  vessel  is  supposed  to  empty  itself  by  an 
orifice  to  which  a  cylindrical  tube  has  been  adapted,  we  can 
still  employ  the  coefficient  (0.81),  provided  we  only  consider 
the  circumstances  of  discharge  previous  to  the  arrival  of  the 
upper  surface  of  the  fluid  at  such  a  level  that  the  tunnel 
begins  to  be  formed  above  the  orifice. 

773.  By  adapting  tubes  of  diflferent  forms  to  an  orifice 
pierced  in  the  side  or  bottom  of  a  vessel,  the  quantity  of  fluid 
discharged  is  generally  found  to  be  more  or  less  increased. 

The  following  table  presents  a  view  of  the  relative  quan- 
tities discharged  in  some  of  the  simplest  cases. 

1°.  Theoretical  discharge  in  a  given  time  through 

an  orifice  pierced  in  a  thin  plate     -    -    -    -     1.00 
2°.  Actual  discharge  in  the  same  time  through  the 

same  orifice 0.62 

3°.  Discharge  through  a  cylindrical  tube,  whose 

length  is  equal   to  two  diameters  of   the 

orifice 0.81 


MOTION    OF   WATER    IN    PIPES.  429 

4°.  Discharge  through  a  conical  tube  having  the 
form  of  the  contracted  vein,  the  larger  base 
being  regarded  as  the  orifice 0.62 

5°.  Discharge  through  the  same  tube,  regarding  the 

smaller  base  as  the  orifice 1.00 


Of  the  Motion  of  Water  in  Pipes. 

774.  Let  AB  {Fig.  254)  represent  a  cylindrical  pipe,  by- 
means  of  which  the  water  contained  in  the  reservoir  R  is 
transferred  to  the  reservoir  R',  and  let  it  be  supposed  that 
the  current  has  assumed  a  uniform  motion  :  it  is  proposed  to 
investigate  a  formula  by  means  of  which  the  quantity  of 
water  delivered  at  the  point  B,  in  a  given  time,  may  be 
estimated. 

Let  CC'D'D  represent  an  elementary  stratum  of  the  fluid 
included  between  two  consecutive  transverse  sections.  Then, 
since  the  motion  is  supposed  to  have  become  uniform,  the 
forces  which  tend  to  accelerate  the  motion  of  the  element  CD' 
must  be  precisely  equal  to  those  which  are  exerted  upon  the 
element  in  a  contrary  direction.  The  force  exerted  upon 
CD',  urging  it  in  a  direction  from  A  towards  B,  is  the  com- 
ponent of  the  weight  of  this  element,  in  a  direction  parallel 
to  the  axis  of  the  pipe  :  and  the  forces  which  urge  it  in  an 
opposite  direction  are,  1^.  the  difference  of  the  pressures 
exerted  upon  the  faces  CC  and  DD'  ;  and,  2".  the  resist- 
ance arising  from  the  friction  of  the  fluid  against  the  sides  of 
the  pipe. 

775.  lip  denote  the  mean  pressure,  referred  to  the  unit  of 
surface,  in  the  section  CC,  the  corresponding  pressure  in  the 
section  DD'  will  be  expressed  by  p-\-dp,  and  if  we  denote  by 
a  the  area  of  the  transverse  section  of  the  pipe,  the  entire 
pressures  upon  the  sections  CC  and  DD'  will  be  respectively 

adp,     a{p  +  dp). 
These  pressures  being  exerted  in  contrary  directions,  the  ele- 
mentary stratum  CD'  will  be  acted  upon  by  a  force  equal  to 
their  difference  adp. 

The  resistance  arising  from  the  friction  against  the  sides 
of  the  pipe  will  be  directly  proportional  to  the  surface  of  the 


430  HYDRODYNAMICS. 

fluid  in  contact  with  the  pipe,  and  will  likewise  be  dependent 
upon  the  velocity  of  the  current.  Hence,  if  v  denote  the 
velocity,  c  the  circumference  of  the  section,  and  s  the  distance 
of  the  section  CC  from  the  extremity  A,  the  distance  CD  will 
be  expressed  by  ds,  and  the  resistance  experienced  by  the 
element  CD',  in  consequence  of  friction,  will  be 

cds  .<p{v)  ; 
in  which  ç(v)  represents  a  certain  function  of  v,  to  be  ascer- 
tained by  experiment. 

776.  To  obtain  an  expression  for  the  force  which  acts  in 
the  direction  from  A  towards  B,  we  shall  suppose  the  density 
of  water  to  be  equal  to  unity,  and  resolve  the  weight  of  the 
element  which  is  expressed  by  g  .  ads  into  two  components, 
respectively  parallel  and  perpendicular  to  the  axis  of  the  pipe. 
Then  denoting  by  ê  the  angle  included  between  the  axis  and 
the  horizon,  the  component  of  the  weight  parallel  to  the  axis 
will  become  ^ .  sin  ^ .  ads.  But  if  z  represent  the  vertical 
co-ordinate  of  the  point  C  referred  to  A  as  an  origin,  dz  will 
represent  the  difference  of  level  of  the  points  C  and  D  ;  and 
we  shall  have 

—  =sin  Ô,     g .  sin  ô .  ads = g  adz 
ds 

And  since  an  equilibrium  must  subsist  between  this  force  and 

the  forces  exerted  in  an  opposite  direction,  we  have 

gadz  =  ad  J)  +  cds .  <p{v)  ; 
and  by  integration, 

gaz  —  ap-^cs.  <p{v)  +  C. 
To  determine  the  value  of  the  constant  C,  we  suppose  the 
pressure  at  the  origin  A  to  be  equal  to  a  known  quantity  P  : 
we  shall  then  have  p  =  F,  z=0,  s=0;  and  therefore 

C=—aF. 
Eliminating  C  between  these  two  equations,  we  obtain 

gaz=a{p—T)-{-cs.<p{v). 
And  by  taking  the  integral  between  the  limits  5=0,  and 
s=AB=l,  the  entire  length  of  the  tube,  denoting  by  F  the 
pressure  at  the  lower  extremity,  and  by  z'  the  co-ordinate  of 
the  point  B,  there  results 

g.az'=a(P'—F)-\-cl.<p{v) (444). 


MOTION    OF    WATER    IN    PIPES.  431 

777.  It  has  been  found  by  experiment,  that  the  function 
ç>{v)  may  be  expressed  by  two  terms  which  are  respectively 
proportional  to  the  first  and  second  powers  of  the  velocity  : 
thus,  we  shall  have 

ç>{v)=bv-^b'v''  ; 
b  and  b'  representing  constant  quantities. 

This  value  of  <p{v),  being  substituted  in  equation  (444), 
gives 

cl 
but  if  the  diameter  of  the  pipe  be  denoted  by  D,  we  shall 
have 


and  therefore, 


-=iD- 


bv + b'v^  =iD^^'     ^f    ^\ 

Ù 


778.  The  pressure  P  at  the  upper  extremity  of  the  pipe 
may  be  regarded  without  material  error  as  that  due  to  the 
depth  E A  of  the  point  A  below  the  surface  of  the  fluid  in  the 
reservoir  R.  Strictly  speaking,  the  pressure  P  is  somewhat 
less  than  that  due  to  the  depth  EA,  since  these  pressures  be- 
come equal  only  when  the  orifice  is  infinitely  small  (Art.  742)  ; 
but  the  difference  is  inconsiderable  when  the  velocity  of  the 
fluid  is  not  great.  In  like  manner,  the  pressure  at  the  point 
B  may  be  supposed  due  to  the  depth  E'B  of  the  point  B  below 
the  surface  of  the  fluid  in  the  reservoir  R'  :  hence,  if  h  and  k' 
represent  the  respective  depths  EA  and  E'B,  we  shall  have 
(Art.  655)  ^=gh,  V=-gh'  ;  and  by  substitution  we  obtain 

bv  +  6'i;2  r=  iDg _2_ . 

If  we  divide  each  member  of  this  equation  by  g^  and  put, 
for  brevity, 

b  b'         z'—h'+h     J 

g         g  « 

we  shall  obtain 

uv-\-^v*={T>k (445). 

The  values  of  «  and  ^  may  be  regarded  as  known,  since 


432  HYDRODYNAMICS. 

they  result  immediately  from  those  of  a  and  6,  which  are  sup- 
posed to  be  determined  by  observation  ;  and  the  value  of  k 
will  likewise  be  given  when  the  length  of  the  pipe,  the  differ- 
ence of  level  of  its  two  extremities,  and  the  difference  of  the 
pressures  at  those  points  are  previously  given.  Hence,  the 
velocity  v  in  a  pipe  of  a  given  diameter  can  be  readily  cal- 
culated. 

779.  The  numerical  values  of  «  and  /3  have  been  found  by 
Prony  to  be 

«=0.00017,     /3 =0.000106; 

and  the  preceding  equation  therefore  becomes 
0.00017t?  +  0.000106^2  ^  ^DA. 
If  we  neglect  the  first  term,  which  is  generally  admissible 
when  the  velocity  v  is  not  extremely  small,  the  formula  will 
reduce  to 

v=48.56^(DA-). 

780.  Let  Q,  denote  the  quantity  delivered  at  the  point  B 
in  a  second  of  time,  and  t  the  number  3.1416  ;  we  shall  have 

and  by  substituting  this  value  of  v  in  equation  (445),  there 
results 

4a  ,  i6a=   ,T., 

or,  if  we  neglect  the  term  containing  the  first  power  of  v,  and 
make  — -=^'^,  we  shall  obtain 

The  numerical  value  of  —  is  38.12  ;  and  the  formula  there- 
9>' 

fore  reduces  to 

a-38.12^(D^A). 

In  this  investigation  the  dimensions  are  supposed  to  be 
expressed  in  English  feet. 

THE    END. 


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